0,3

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)

M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217.

D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, p. 255-7,276

F. Beukers, Letter to D. Shanks, Mar 13 1984

T. D. Noe, Table of n, a(n) for n=0..100

a(n)=(1/24) * Coefficient x^n in Product_{k=1..inf} (1+x^(2k-1))^(24n).

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

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

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}]

(PARI) a(n)=polcoeff(prod(k=1, (n+1)\2, 1+x^(2*k-1), 1+x*O(x^n))^(24*n), n)/24

(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))}

(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; )}

(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])

Cf. A005149.

A060236 (reduced mod 3).

Adjacent sequences: A005145 A005146 A005147 this_sequence A005149 A005150 A005151

Sequence in context: A049668 A009991 A052463 this_sequence A123798 A104069 A101629

nonn,easy,nice

Simon Plouffe, njas

More terms from Michael Somos, Nov 24 2001