{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "01." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "f:=(n,k)->k*binomial(n,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&9%\" \"\"-%)binomialG6$9$F.F/F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 " 02." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ansatz:=sum(sum(a[i, j]*F(n+j,k+i),i=0..1),j=0..1)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'ansatzG/,**&&%\"aG6$\"\"!F+\"\"\"-%\"FG6$%\"nG%\"kGF,F,*&&F)6$F,F+F,- F.6$F0,&F1F,F,F,F,F,*&&F)6$F+F,F,-F.6$,&F0F,F,F,F1F,F,*&&F)6$F,F,F,-F. 6$F=F7F,F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "03." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sum(sum(a[i,j]*F(n+j,k+i)/F(n,k),i= 0..1),j=0..1)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"aG6$\"\"!F( \"\"\"*(&F&6$F)F(F)-%\"FG6$%\"nG,&%\"kGF)F)F)F)-F.6$F0F2!\"\"F)*(&F&6$ F(F)F)-F.6$,&F0F)F)F)F2F)F3F5F)*(&F&6$F)F)F)-F.6$F;F1F)F3F5F)F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "04." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "SUM:=map(y->normal(expand(y)),sum(sum(a[i,j]*f(n+j,k+ i)/f(n,k),i=0..1),j=0..1))=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SU MG/,*&%\"aG6$\"\"!F*\"\"\"*(,&%\"nGF+%\"kG!\"\"F+&F(6$F+F*F+F/F0F+*(&F (6$F*F+F+,&F.F+F+F+F+,(F.F+F+F+F/F0F0F+*(F6F+&F(6$F+F+F+F/F0F+F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "05." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "SUM:=numer(normal(lhs(SUM)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SUMG,@*(&%\"aG6$\"\"!F*\"\"\"%\"kGF+%\"nGF+F+*&F'F+F ,F+F+*&F'F+)F,\"\"#F+!\"\"*&&F(6$F+F*F+)F-F1F+F+*&F4F+F-F+F+**F1F+F4F+ F-F+F,F+F2*&F4F+F,F+F2*&F4F+F0F+F+*(&F(6$F*F+F+F,F+F-F+F+*&F " 0 "" {MPLTEXT 1 0 43 "liste:=\{coeff(SUM,k,i) $ i=0..degree(SUM)\};" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&listeG<&\"\"!,&&%\"aG6$\"\"\"F&F+&F )6$F&F&!\"\",2F,F+*(\"\"#F+F(F+%\"nGF+F.F(F.*&&F)6$F&F+F+F2F+F+F4F+&F) 6$F+F+F.*&F6F+F2F+F.*&F,F+F2F+F+,,F6F+*&F6F+)F2F1F+F+*&F(F+F " 0 "" {MPLTEXT 1 0 52 "loesung:=solve(liste,\{a[0, 0],a[1,0],a[0,1],a[1,1]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(loes ungG<&/&%\"aG6$\"\"!F*,$*(%\"nG!\"\",&F-\"\"\"F0F0F0&F(6$F0F0F0F./&F(6 $F0F*F+/F1F1/&F(6$F*F0F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "08." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "RE:=numer(normal(subs(loes ung,lhs(ansatz))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,$*&&%\"a G6$\"\"\"F*F*,,*&-%\"FG6$%\"nG%\"kGF*F0F*F*F-F**&-F.6$F0,&F1F*F*F*F*F0 F*F*F3F**&-F.6$,&F0F*F*F*F5F*F0F*!\"\"F*F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "09." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "kfreieR E:=proc(f,k,kmax,n,nmax,F)\noption remember;\nlocal variablen,SUM,i,j, ansatz,liste,loesung,RE;\nvariablen:=\{seq(seq(a[i,j],j=0..kmax),i=0.. nmax)\};\nansatz:=sum(sum(a[i,j]*F(n+i,k+j),j=0..kmax),i=0..nmax);\nSU M:=simplify(expand(sum(sum(a[i,j]*subs(\{k=k+j,n=n+i\},f)/f,i=0..nmax) ,j=0..kmax)));\nSUM:=collect(numer(normal(SUM)),k);\nliste:=\{seq(coef f(SUM,k,i),i=0..degree(SUM,k))\};\nloesung:=solve(liste,variablen);\ni f evalb(subs(loesung,variablen)=\{0\})\n then return \"Es existiert k eine solche Rekursion!\"\nend if;\nRE:=numer(normal(subs(loesung,ansat z)));\nRE:=collect(RE,F);\nRE:=map(factor,RE);\nRE=0\nend proc:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "10." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "RE:=kfreieRE(k*binomial(n,k),k,1,n,1,G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(*(,&%\"nG\"\"\"F*F*F*&%\"aG6$F*F*F*-% \"GG6$F)%\"kGF*!\"\"*(F(F*F+F*-F/6$F),&F1F*F*F*F*F2*(F+F*-F/6$F(F6F*F) F*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "11." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "kfreieRE(binomial(n,k),k,1,n,1,G);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&&%\"aG6$\"\"!\"\"\"F*-%\"GG6$%\"n G%\"kGF*F**&F&F*-F,6$F.,&F/F*F*F*F*F**&F&F*-F,6$,&F.F*F*F*F3F*!\"\"F) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "12." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "kfreieRE((binomial(n,k))^2,k,2,n,2,G);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*(,&%\"nG\"\"\"F(F(F(&%\"aG6$F(\"\"#F(-% \"GG6$F'%\"kGF(!\"\"**F,F(F&F(F)F(-F.6$F',&F0F(F(F(F(F(*(F)F(-F.6$F&F5 F(,&*&F,F(F'F(F(\"\"$F(F(F(*(F&F(F)F(-F.6$F',&F0F(F,F(F(F1*(F)F(-F.6$F &F?F(F9F(F(*(,&F'F(F,F(F(F)F(-F.6$FDF?F(F1\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "13." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "RE: =applyrule(G(n::anything,k::anything)=S(n),lhs(RE));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,&**\"\"#\"\"\",&%\"nGF(F(F(F(&%\"aG6$F(F(F(-% \"SG6#F*F(!\"\"*(F+F(-F/6#F)F(F*F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "14." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "FasenmyerRE:= proc(f,k,kmax,n,nmax)\nlocal RE;\nRE:=kfreieRE(f,k,kmax,n,nmax,G);\nif RE=\"Es existiert keine solche Rekursion!\"\n then return \"Es exist iert keine solche Rekursion!\"\nend if;\nRE:=applyrule(G(n::anything,k ::anything)=S(n),lhs(RE))=0;\ncollect(subs(\{seq(seq(a[i,j]=1,i=0..nma x),j=0..kmax)\},RE),S,factor);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "15." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "RE:=Fase nmyerRE(k*binomial(n,k),k,1,n,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#REG/,&*&,&\"\"#!\"\"*&F)\"\"\"%\"nGF,F*F,-%\"SG6#F-F,F,*&-F/6#,&F-F, F,F,F,F-F,F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "16." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "FasenmyerRE(binomial(n,k),k, 1,n,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"#\"\"\"-%\"SG6#%\" nGF'F'-F)6#,&F+F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "17." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "FasenmyerRE(binomia l(n,k)^2,k,2,n,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&\"\"%\" \"\"%\"nGF)F)\"\"'F)F)-%\"SG6#,&F*F)F)F)F)F)*&,&F*!\"\"\"\"#F2F)-F-6#, &F*F)F3F)F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "18." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FasenmyerRE((-1)^k*binomial( n,k)^2,k,2,n,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&\"\"%\"\" \"%\"nGF)!\"\"F(F+F)-%\"SG6#F*F)F)*&,&F*F+\"\"#F+F)-F-6#,&F*F)F1F)F)F) \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "19." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "FasenmyerRE(binomial(n,k)*binomial(-n-1,k)*(( 1-x)/2)^k,k,1,n,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\" \"\"F(F(F(-%\"SG6#F'F(F(*(%\"xGF(,&*&\"\"#F(F'F(F(\"\"$F(F(-F*6#F&F(! \"\"*&,&F'F(F0F(F(-F*6#F6F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "20." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "FasenmyerRE(1/ 2^n*(-1)^k*binomial(n,k)*binomial(2*n-2*k,n)*x^(n-2*k),k,1,n,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"F(F(F(-%\"SG6#F'F(F( *(%\"xGF(,&*&\"\"#F(F'F(F(\"\"$F(F(-F*6#F&F(!\"\"*&,&F'F(F0F(F(-F*6#F6 F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }