{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 }{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 }} {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);" }}}{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;" }}}{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;" }}}{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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "05." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "SUM:=numer(normal(lhs(SUM))) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "06." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "liste:=\{coeff(SUM,k,i) $ i=0..degree(SUM)\};" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "07." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "loesung:=solve(liste,\{a[0,0],a[1,0],a[0,1],a[1,1]\}) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "08." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "RE:=numer(normal(subs(loesung,lhs(ansatz))));" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "09." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "kfreieRE:=proc(f,k,kmax,n,nmax,F)\noption remember; \nlocal variablen,SUM,i,j,ansatz,liste,loesung,RE;\nvariablen:=\{seq(s eq(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);\nSUM:=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(SU M)),k);\nliste:=\{seq(coeff(SUM,k,i),i=0..degree(SUM,k))\};\nloesung:= solve(liste,variablen);\nif evalb(subs(loesung,variablen)=\{0\})\n th en return \"Es existiert keine solche Rekursion!\"\nend if;\nRE:=numer (normal(subs(loesung,ansatz)));\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);" }}}{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); " }}}{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);" }}}{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));" }}} {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:=k freieRE(f,k,kmax,n,nmax,G);\nif RE=\"Es existiert keine solche Rekursi on!\"\n then return \"Es existiert keine solche Rekursion!\"\nend if; \nRE:=applyrule(G(n::anything,k::anything)=S(n),lhs(RE))=0;\ncollect(s ubs(\{seq(seq(a[i,j]=1,i=0..nmax),j=0..kmax)\},RE),S,factor);\nend pro c:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "15." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "RE:=FasenmyerRE(k*binomial(n,k),k,1,n,1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "16." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "FasenmyerRE(binomial(n,k),k,1,n,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "17." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "FasenmyerRE(binomial(n,k)^2,k,2,n,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "18." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Fasenmye rRE((-1)^k*binomial(n,k)^2,k,2,n,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "19." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "FasenmyerRE(bi nomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,1,n,2);" }}}{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);" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }