Search: id:A005148
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%I A005148 M5290
%S A005148 0,1,47,2488,138799,7976456,467232200,27736348480,1662803271215,
%T A005148 100442427373480,6103747246289272,372725876150863808,
%U A005148 22852464771010647496,1405886026610765892544,86741060172969340021952
%N A005148 Sequence of coefficients arising in connection with a rapidly converging 
               series for Pi.
%C A005148 The paper by Newman and Shanks has an appendix by Don Zagier which eventually 
               leads to an efficient recursive algorithm for the series itself, 
               whereas the main paper treats each term in isolation, which is enormously 
               slower. Using Zagier's appendix one may compute 1000 terms in 25 
               seconds running Pari-GP on a 500MHz Alpha. - D.Broadhurst(AT)open.ac.uk, 
               Jun 17 2002 (see second version of PARI code here)
%D A005148 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005148 M. Newman and D. Shanks, On a sequence arising in series for pi, Math. 
               Comp., 42 (1984), 199-217.
%D A005148 D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 
               1985, p. 255-7,276
%D A005148 F. Beukers, Letter to D. Shanks, Mar 13 1984
%H A005148 T. D. Noe, <a href="b005148.txt">Table of n, a(n) for n=0..100</a>
%H A005148 <a href="Sindx_Ph.html#Pi314">Index entries for sequences related to 
               the number Pi</a>
%F A005148 a(n)=(1/24) * Coefficient x^n in Product_{k=1..inf} (1+x^(2k-1))^(24n).
%F A005148 Asymptotically (D. Zagier): a(n) = C*(64^n)/sqrt(n)*(1-a/n+b/n^2+...) 
               with C = sqrt(Pi)/12 *Gamma(3/4)^2/Gamma(1/4)^2 = 0.0168732651.... 
               a=6*Gamma(3/4)^4/Gamma(1/4)^4 = 0.07830067... b=60*Gamma(3/4)^8/Gamma(1/
               4)^8 - 1/128 = 0.002405668.... - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 22 2002
%F A005148 Sum_i^n binom(2n-2i, n-i)^3 a(i) = 1/24 binom(2n, n)(16^n-binom(2n, n)^2) 
               (Shanks and Beukers). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Oct 24 2002
%t A005148 a[n_] := a[n]=(Binomial[2n, n](16^n-Binomial[2n, n]^2))/24-Sum[Binomial[2n-2i, 
               n-i]^3a[i], {i, 0, n-1}]
%o A005148 (PARI) a(n)=polcoeff(prod(k=1,(n+1)\2,1+x^(2*k-1),1+x*O(x^n))^(24*n), 
               n)/24
%o A005148 (PARI) {nt=1000; a=[1]; b=[1]; d=1; e=0; g=0; print(1); for(n=2,nt, c=48*(a[n-1]+g)+128*(d-32*e); 
               e=d; d=c; i=(n-1)\2; g=12*if(n%2==0,a[n/2]^2)+24*sum(j=1,i,a[j]*a[n-j]); 
               h=12*if(n%2==0,b[n/2]^2)+24*sum(j=1,i,b[j]*b[n-j]); f=(c+5*h)/n^2-g; 
               a=concat(a,f); b=concat(b,n*f); print(f))}
%o A005148 (PARI) {a(n)=if(n<1,0,va[n])} {b(n)=n*a(n)} {doit(nt)= local(c,d,e,g); 
               va=vector(nt); va[1]=1; d=1; e=0; g=0; for(n=2,nt, c=48*(a(n-1)+g)+128*(d-32*e); 
               e=d; d=c; g=12*if(n%2==0,a(n/2)^2)+24*sum(j=1,(n-1)\2,a(j)*a(n-j)); 
               va[n]=(c+5*(12*if(n%2==0,b(n/2)^2)+24*sum(j=1,(n-1)\2,b(j)*b(n-j))))/
               n^2-g; )}
%o A005148 (PARI) a(n)=local(an, cb); if(n<1,0,an=cb=vector(n,i,binomial(2*i,i)); 
               an[1]=1; for(j=2,n,an[j]=(cb[j]*16^j-cb[j]^3)/24-sum(i=1,j-1,cb[j-i]^3*an[i])); 
               an[n])
%Y A005148 Cf. A005149.
%Y A005148 A060236 (reduced mod 3).
%Y A005148 Sequence in context: A049668 A009991 A052463 this_sequence A123798 A104069 
               A141797
%Y A005148 Adjacent sequences: A005145 A005146 A005147 this_sequence A005149 A005150 
               A005151
%K A005148 nonn,easy,nice
%O A005148 0,3
%A A005148 Simon Plouffe, N. J. A. Sloane (njas(AT)research.att.com).
%E A005148 More terms from Michael Somos, Nov 24 2001

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