{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "The Euler numbers" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "The Euler numbers" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 13 "Euler numbers" } {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon[n];" "6#&%(epsilonG6#%\"nG" } {TEXT -1 73 " are obtained from the coefficients in the Maclaurin seri es expansion of " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "sech*x = 1/(cosh*x);" "6#/*&%%sechG\"\"\"%\"xGF&*&F&F&*&%%coshGF&F'F& !\"\"" }{XPPEDIT 18 0 "``= 2/(exp(x)+exp(-x))" "6#/%!G*&\"\"#\"\"\",&- %$expG6#%\"xGF'-F*6#,$F,!\"\"F'F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "using the formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sech*x = Sum(epsilon[n]/n!*x^n,n = 0 .. infinity);" "6# /*&%%sechG\"\"\"%\"xGF&-%$SumG6$*(&%(epsilonG6#%\"nGF&-%*factorialG6#F /!\"\")F'F/F&/F/;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "taylor(sech( x),x,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!#!\"\" \"\"#F)#\"\"&\"#C\"\"%#!#h\"$?(\"\"'#\"$x#\"%k!)\"\")#!&@0&\"(+)GO\"#5 #\"'`0a\")?.!e*\"#7#!*\")4O*>\",+7Hyr)\"#9-%\"OG6#F%\"#:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For example," }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon[0]=1" "6#/&%(ep silonG6#\"\"!\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon[2]/2!=-1/2" "6#/*&&%(epsilonG6#\"\"#\"\"\"-%*f actorialG6#F(!\"\",$*&F)F)F(F-F-" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 " epsilon(2)=-1" "6#/-%(epsilonG6#\"\"#,$\"\"\"!\"\"" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon[4]/4!=5/24" "6 #/*&&%(epsilonG6#\"\"%\"\"\"-%*factorialG6#F(!\"\"*&\"\"&F)\"#CF-" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "epsilon(4)=5" "6#/-%(epsilonG6#\"\" %\"\"&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "epsilon[6]/6! = -61/720;" "6#/*&&%(epsilonG6#\"\"'\"\"\"-%*facto rialG6#F(!\"\",$*&\"#hF)\"$?(F-F-" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "epsilon[6] = -61;" "6#/&%(epsilonG6#\"\"',$\"#h!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The E uler numbers can be obtained using the Maple procedure " }{TEXT 0 5 "e uler" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "euler(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The previous truncated Maclaurin series for " }{XPPEDIT 18 0 "sech *x;" "6#*&%%sechG\"\"\"%\"xGF%" }{TEXT -1 33 " can also be obtained as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "alias(epsilon=euler):\nSum(epsilon(n)/n!*x^n,n=0..14) ;\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(-%(epsilonG 6#%\"nG\"\"\"-%*factorialGF)!\"\")%\"xGF*F+/F*;\"\"!\"#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,2\"\"\"F$*&#F$\"\"#F$*$)%\"xGF'F$F$!\"\"*&#\"\" &\"#CF$*$)F*\"\"%F$F$F$*&#\"#h\"$?(F$*$)F*\"\"'F$F$F+*&#\"$x#\"%k!)F$* $)F*\"\")F$F$F$*&#\"&@0&\"(+)GOF$*$)F*\"#5F$F$F+*&#\"'`0a\")?.!e*F$*$) F*\"#7F$F$F$*&#\"*\")4O*>\",+7Hyr)F$*$)F*\"#9F$F$F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " sech*x;" "6#*&%%sechG\"\"\"%\"xGF%" }{TEXT -1 40 " is an even function , the Euler numbers " }{XPPEDIT 18 0 "epsilon[n];" "6#&%(epsilonG6#%\" nG" }{TEXT -1 5 " for " }{TEXT 262 1 "n" }{TEXT -1 18 " odd are all ze ro." }}{PARA 0 "" 0 "" {TEXT -1 79 "This means that we can also obtain the previous truncated Maclaurin series for " }{XPPEDIT 18 0 "sech*x; " "6#*&%%sechG\"\"\"%\"xGF%" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum(ep silon(2*n)/(2*n)!*x^(2*n),n=0..7);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(-%(epsilonG6#,$*&\"\"#\"\"\"%\"nGF-F-F--%*fa ctorialGF)!\"\")%\"xGF*F-/F.;\"\"!\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\"\"\"F$*&#F$\"\"#F$*$)%\"xGF'F$F$!\"\"*&#\"\"&\"#CF$*$)F*\"\" %F$F$F$*&#\"#h\"$?(F$*$)F*\"\"'F$F$F+*&#\"$x#\"%k!)F$*$)F*\"\")F$F$F$* &#\"&@0&\"(+)GOF$*$)F*\"#5F$F$F+*&#\"'`0a\")?.!e*F$*$)F*\"#7F$F$F$*&# \"*\")4O*>\",+7Hyr)F$*$)F*\"#9F$F$F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 13 "The formula: " }{XPPEDIT 18 0 "Sum(matrix([[2*n], \+ [k]])*epsilon[2*k],k = 0 .. p) = 0;" "6#/-%$SumG6$*&-%'matrixG6#7$7#*& \"\"#\"\"\"%\"nGF/7#%\"kGF/&%(epsilonG6#*&F.F/F2F/F//F2;\"\"!%\"pGF9" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/( cosh*x) = x/(1+x^2/2!+x^4/4!+x^6/6!+`. . . .`);" "6#/*&\"\"\"F%*&%%cos hGF%%\"xGF%!\"\"*&F(F%,,F%F%*&F(\"\"#-%*factorialG6#F-F)F%*&F(\"\"%-F/ 6#F2F)F%*&F(\"\"'-F/6#F6F)F%%(.~.~.~.GF%F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(1+x^2/2!+x^4/4!+x^6 /6!+` . . . `);" "6#/%!G*&\"\"\"F&,,F&F&*&%\"xG\"\"#-%*factorialG6#F*! \"\"F&*&F)\"\"%-F,6#F0F.F&*&F)\"\"'-F,6#F4F.F&%(~.~.~.~GF&F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(epsilon[0]+epsilon[2]*x^2/2!+epsilon [4]*x^4/4!+epsilon[6]*x^6/6!+` . . . `)*(1+x^2/2!+x^4/4!+x^6/6!+` . . \+ . `) = 1;" "6#/*&,,&%(epsilonG6#\"\"!\"\"\"*(&F'6#\"\"#F**$%\"xGF.F*-% *factorialG6#F.!\"\"F**(&F'6#\"\"%F**$F0F8F*-F26#F8F4F**(&F'6#\"\"'F** $F0F?F*-F26#F?F4F*%(~.~.~.~GF*F*,,F*F**&F0F.-F26#F.F4F**&F0F8-F26#F8F4 F**&F0F?-F26#F?F4F*FCF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Comparing coefficients of power s of x on the left and right sides we have" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^0" "6#*$%\"xG\"\"!" }{TEXT -1 5 ": " } {XPPEDIT 18 0 "epsilon[0] = 1;" "6#/&%(epsilonG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2" "6#*$% \"xG\"\"#" }{TEXT -1 5 ": " }{XPPEDIT 18 0 "epsilon[0]/(2!*0!)+epsi lon[2]/(0!*2!) = 0;" "6#/,&*&&%(epsilonG6#\"\"!\"\"\"*&-%*factorialG6# \"\"#F*-F-6#F)F*!\"\"F**&&F'6#F/F**&-F-6#F)F*-F-6#F/F*F2F*F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4" "6#*$% \"xG\"\"%" }{TEXT -1 6 ": " }{XPPEDIT 18 0 "epsilon[0]/(4!*0!)+eps ilon[2]/(2!*2!)+epsilon[4]/(0!*4!);" "6#,(*&&%(epsilonG6#\"\"!\"\"\"*& -%*factorialG6#\"\"%F)-F,6#F(F)!\"\"F)*&&F&6#\"\"#F)*&-F,6#F5F)-F,6#F5 F)F1F)*&&F&6#F.F)*&-F,6#F(F)-F,6#F.F)F1F)" }{TEXT -1 5 " = 0 " }} {PARA 0 "" 0 "" {TEXT -1 30 " :" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x^(2*n);" "6#)%\"xG*&\"\"#\"\"\"%\"nGF'" } {TEXT -1 8 ": " }{XPPEDIT 18 0 "epsilon[0]/((2*n)!*0!)+epsilon[2 ]/((2*n-2)!*2!)+epsilon[4]/((2*n-4)!*4!)+` . . . `+epsilon[2*n]/(0!*(2 *n)!);" "6#,,*&&%(epsilonG6#\"\"!\"\"\"*&-%*factorialG6#*&\"\"#F)%\"nG F)F)-F,6#F(F)!\"\"F)*&&F&6#F/F)*&-F,6#,&*&F/F)F0F)F)F/F3F)-F,6#F/F)F3F )*&&F&6#\"\"%F)*&-F,6#,&*&F/F)F0F)F)FAF3F)-F,6#FAF)F3F)%(~.~.~.~GF)*&& F&6#*&F/F)F0F)F)*&-F,6#F(F)-F,6#*&F/F)F0F)F)F3F)" }{TEXT -1 6 " = 0 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " Mult iplying the last equation by " }{XPPEDIT 18 0 "(2*n)!;" "6#-%*factoria lG6#*&\"\"#\"\"\"%\"nGF(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2*n], [0]])*epsilon[0]+matrix( [[2*n], [2]])*epsilon[1]+matrix([[2*n], [4]])*epsilon[2]+` . . . `+mat rix([[2*n], [2*n]])*epsilon[2*n] = 0;" "6#/,,*&-%'matrixG6#7$7#*&\"\"# \"\"\"%\"nGF-7#\"\"!F-&%(epsilonG6#F0F-F-*&-F'6#7$7#*&F,F-F.F-7#F,F-&F 26#F-F-F-*&-F'6#7$7#*&F,F-F.F-7#\"\"%F-&F26#F,F-F-%(~.~.~.~GF-*&-F'6#7 $7#*&F,F-F.F-7#*&F,F-F.F-F-&F26#*&F,F-F.F-F-F-F0" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(matrix([[2*n], [2*k]])*epsilon[2*k],k = 0 .. n) = 0 " "6#/-%$SumG6$*&-%'matrixG6#7$7#*&\"\"#\"\"\"%\"nGF/7#*&F.F/%\"kGF/F/ &%(epsilonG6#*&F.F/F3F/F//F3;\"\"!F0F:" }{TEXT -1 15 " ------- (i), \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 11 "___________" } {TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "matrix([[ 2*n], [2*k]]);" "6#-%'matrixG6#7$7#*&\"\"#\"\"\"%\"nGF*7#*&F)F*%\"kGF* " }{TEXT -1 35 " denotes the binomial coefficient. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 14 "If we replace " }{XPPEDIT 18 0 "epsilon[ k];" "6#&%(epsilonG6#%\"kG" }{TEXT -1 14 " by the power " }{XPPEDIT 18 0 "epsilon^k;" "6#)%(epsilonG%\"kG" }{TEXT -1 49 ", formula (i) gi ves rise to the symbolic formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(matrix([[2*n], [2 *k]])*epsilon^(2*k),k = 0 .. p) = 0;" "6#/-%$SumG6$*&-%'matrixG6#7$7#* &\"\"#\"\"\"%\"nGF/7#*&F.F/%\"kGF/F/)%(epsilonG*&F.F/F3F/F//F3;\"\"!% \"pGF9" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 34 "which can be e xpressed in the form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1+epsilon)^(2*n)-(1-epsilon)^(2*n) = 0;" "6#/,&),&\"\"\"F'%(epsilo nGF'*&\"\"#F'%\"nGF'F'),&F'F'F(!\"\"*&F*F'F+F'F.\"\"!" }{TEXT -1 1 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 17 "_________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 " " }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Calculation of the Euler numbers " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 11 "The formula" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "Sum(matrix([[2*n], [2*k]])*epsilon[2*k],k = 0 .. n) = 0 ;" "6#/-%$SumG6$*&-%'matrixG6#7$7#*&\"\"#\"\"\"%\"nGF/7#*&F.F/%\"kGF/F /&%(epsilonG6#*&F.F/F3F/F//F3;\"\"!F0F:" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 44 "can be used to calculate the Euler nu mbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "n = 4;" "6#/%\"nG\"\"%" } {TEXT -1 8 " we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon[0]+28*epsilon[2]+70*epsilon[4]+28*epsilon[6]+epsilon[8] = 0 ;" "6#/,,&%(epsilonG6#\"\"!\"\"\"*&\"#GF)&F&6#\"\"#F)F)*&\"#qF)&F&6#\" \"%F)F)*&F+F)&F&6#\"\"'F)F)&F&6#\"\")F)F(" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "alias(epsilon=epsilon): n := 4:\nad d(binomial(2*n,2*k)*epsilon[2*k],k=0..n);\nn := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%(epsilonG6#\"\"!\"\"\"*&\"#GF(&F%6#\"\"#F(F(*&\" #qF(&F%6#\"\"%F(F(*&F*F(&F%6#\"\"'F(F(&F%6#\"\")F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "This means that we can compute " }{XPPEDIT 18 0 "epsilon[8];" "6#&%(epsilonG6#\"\")" }{TEXT -1 17 " from the values \+ " }{XPPEDIT 18 0 "epsilon[0] = 1,epsilon[2] = -1,epsilon[4] = 5,epsilo n[6] = -61;" "6&/&%(epsilonG6#\"\"!\"\"\"/&F%6#\"\"#,$F(!\"\"/&F%6#\" \"%\"\"&/&F%6#\"\"',$\"#hF." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{XPPEDIT 18 0 "1-28+350-1708+epsilon[8] = 0;" "6#/,,\" \"\"F%\"#G!\"\"\"$]$F%\"%3 " 0 "" {MPLTEXT 1 0 88 "alias(epsilon=epsilon):\nnextepsilon := n->-sum(epsil on[2*k]*binomial(n,2*k),k=0..n/2-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%,nextepsilonGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,$-%$sumG6$*&&%(ep silonG6#,$*&\"\"#\"\"\"%\"kGF7F7F7-%)binomialG6$9$F4F7/F8;\"\"!,&*&#F7 F6F7F " 0 "" {MPLTEXT 1 0 79 "epsilon[0] := 1;\nfor i from 1 to 10 do epsi lon[2*i] := nextepsilon(2*i) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%(epsilonG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(eps ilonG6#\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\" \"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\"\"'!#h" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\"\")\"%&Q\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\"#5!&@0&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\"#7\"(lFq#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6#\"#9!*\")4O*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%(epsilonG6#\"#;\",X@^\"R>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(eps ilonG6#\"#=!.Tanz[S#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(epsilonG6# \"#?\"0DvB)=r.P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "euler(20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"0DvB)=r.P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Procedu res for computing the Euler numbers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 90 "This section contains the code for two M aple procedures which calculate the Euler numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "1st Euler number procedure" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "euler_1" }{TEXT -1 25 " below uses the formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "epsilon[2*n] = -`` " "6#/&%(epsilonG6#*&\"\"#\"\"\"%\"n GF),$%!G!\"\"" }{XPPEDIT 18 0 "Sum(matrix([[2*n], [2*k]])*epsilon[2*k] ,k = 0 .. n-1);" "6#-%$SumG6$*&-%'matrixG6#7$7#*&\"\"#\"\"\"%\"nGF.7#* &F-F.%\"kGF.F.&%(epsilonG6#*&F-F.F2F.F./F2;\"\"!,&F/F.F.!\"\"" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 71 "where the binomial coefficie nts are computed using the Maple procedure " }{TEXT 0 8 "binomial" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 87 "This is quite efficient \+ because, when computing the sequence of binomial coefficients, " } {TEXT 0 8 "binomial" }{TEXT -1 132 " uses the most recently computed b inomial coefficient to compute the next one, with a minimum number of \+ extra arithmetic operations." }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 7 "euler_1" }{TEXT -1 214 " uses the \"remember\" opti on to save all values computed in the current Maple session in a remem ber table. It operates recursively, bulding up all lower Euler numbers not previously computed in its remember table. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 488 "euler_1 := \+ proc(n::algebraic)\n local e,k;\n option remember;\n if nargs<>1 then \n error \"expecting 1 argument, got %1\",nargs; \n end i f;\n if type(n,'integer') and n>=0 then\n if irem(n,2)<>0 then \+ 0\n elif n=0 then 1\n else\n e := 0;\n for k from 0 to n/2-1 do\n e := e+euler_1(2*k)*binomial(n,2*k)\n end do:\n -e;\n end if;\n elif type(p,'constan t') then error \"invalid argument\"\n else 'euler_1'(n)\n end if; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "euler_1(20);\neuler(20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"0DvB)=r.P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"0DvB)= r.P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "T he remember table can be viewed as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "op(4,op(euler_1)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7-/\"\"!\"\"\"/\"\"#!\" \"/\"\"%\"\"&/\"\"'!#h/\"\")\"%&Q\"/\"#5!&@0&/\"#7\"(lFq#/\"#9!*\")4O* >/\"#;\",X@^\"R>/\"#=!.Tanz[S#/\"#?\"0DvB)=r.P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "2nd Euler number procedure" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 7 "euler_2" }{TEXT -1 70 " given below has been extracted from the Maple code for the procedure " }{TEXT 0 5 "euler" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 123 "This procedure differs from the firs t procedure in that the binomial coefficients are computed without usi ng the procedure " }{TEXT 0 8 "binomial" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 542 "euler_ 2 := proc(n::algebraic)\n local t,e,i;\n option remember;\n if n args<>1 then \n error \"expecting 1 argument, got %1\",nargs; \n \+ end if;\n if type(n,'integer') and 0<=n then\n if irem(n,2)<> 0 then 0\n elif n=0 then 1\n else\n t := 1;\n \+ e := 1;\n for i from 2 by 2 to n-2 do\n t := iquo (t*(n+1-i)*(n+2-i),(i-1)*i);\n e := e+t*euler_2(i)\n \+ end do;\n -e\n end if\n elif type(n,'constant') then \+ error \"invalid arguments\"\n else 'euler_2(n)'\n end if\nend proc :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "euler_2(20);\neuler(20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"0DvB)=r.P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"0DvB)= r.P" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }