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 A005148 Sequence of coefficients arising in connection with a rapidly converging series for Pi. (Formerly M5290) 10

%I M5290

%S 0,1,47,2488,138799,7976456,467232200,27736348480,1662803271215,

%T 100442427373480,6103747246289272,372725876150863808,

%U 22852464771010647496,1405886026610765892544,86741060172969340021952

%N Sequence of coefficients arising in connection with a rapidly converging series for Pi.

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

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

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005148/b005148.txt">Table of n, a(n) for n=0..100</a>

%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>

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

%F 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_, Jun 22 2002

%F 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_, Oct 24 2002

%t 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 (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 (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 (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 (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 Cf. A005149.

%Y A060236 (reduced mod 3).

%K nonn,easy,nice,changed

%O 0,3

%A _Simon Plouffe_ and _N. J. A. Sloane_.

%E More terms from Michael Somos, Nov 24 2001

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