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A005148 Sequence of coefficients arising in connection with a rapidly converging series for Pi.

%I A005148 M5290 #80 Aug 02 2023 18:57:05
%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. - _David Broadhurst_, Jun 17 2002 (see second version of PARI code here)
%C A005148 Conjecture: The following 2 definitions give the same sequence: (1) numbers k such that 8^m is the highest power of 2 dividing a(k), and (2) numbers k such that k has exactly (m+1) 1's in its binary representation. A018900 is the special case m=1. - _Benoit Cloitre_, Jun 22 2002, edited by _Hugo Pfoertner_, Aug 21 2021
%C A005148 Conjecture: There are polynomials P_k(x) such that P_k(m) = the constant term of j_m(tau)^k where j_m is modular for the Hecke group G(lambda_m), j_3 is the Klein invariant j with constant term 744, and P_k(x) = a(k+1) times a product of monic polynomials. - _Barry Brent_, Nov 25 2022
%D A005148 F. Beukers, Letter to D. Shanks, Mar 13 1984
%D A005148 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195; see Exercise 6(a).
%D A005148 D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, p. 255-7,276
%D A005148 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005148 Alois P. Heinz, <a href="/A005148/b005148.txt">Table of n, a(n) for n = 0..555</a> (first 101 terms from T. D. Noe)
%H A005148 Barry Brent, <a href="https://github.com/barry314159a/NewmanShanks">Folder : "current draft"</a>
%H A005148 Barry Brent, <a href="https://arxiv.org/abs/2212.12515">On the constant terms of certain meromorphic modular forms for Hecke groups</a>, arXiv:2212.12515 [math.NT], 2022.
%H A005148 Barry Brent, <a href="https://doi.org/10.20944/preprints202306.1164.v6">On the Constant Terms of Certain Laurent Series</a>, Preprints (2023) 2023061164.
%H A005148 M. Newman and D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1984-0725996-9">On a sequence arising in series for pi</a>, Math. Comp., 42 (1984), 199-217.
%H A005148 D. Shanks, <a href="/A005148/a005148.pdf">Letter to N. J. A. Sloane</a>, date unknown.  Also includes some notes from N. J. A. Sloane.
%H A005148 <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>
%F A005148 a(n) = (1/24) * coefficient of x^n in Product_{k>=1} (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 = 0.078300067..., b = 60*Gamma(3/4)^8/Gamma(1/4)^8 - 1/128 = 0.002405668.... - _Benoit Cloitre_, Jun 22 2002; numerical value of constant "a" corrected by _Vaclav Kotesovec_, Jul 28 2013
%F A005148 Alternative expressions for these constants: C = Pi^(5/2)/(6*Gamma(1/4)^4), a = 24*Pi^4/Gamma(1/4)^8, b = 960*Pi^8/Gamma(1/4)^16 - 1/128. - _Vaclav Kotesovec_, Jul 28 2013
%F A005148 A076657(n) = Sum_{i=0..n} binomial(2*n-2*i, n-i)^3 a(i) = (1/24)*binomial(2*n, n)*(16^n-binomial(2*n, n)^2) (Shanks and Beukers). - _Ralf Stephan_, Oct 24 2002
%F A005148 Expansion of ((Pi / (2 K(q)))^2 / (1 - 2*k(q)^2) - 1) / 24 in powers of (k'(q) * k(q) / 4)^2. [Borwein and Borwein, 6(a)(i)] - _Michael Somos_, Jul 06 2014
%F A005148 Expansion of ((Pi / (2 K(q)))^2 / (1 + k(q)^2) - 1) / 24 in powers of (k'(q)^-2 * k(q) / 4)^2. [Borwein and Borwein, 6(a)(ii)] - _Michael Somos_, Jul 06 2014
%e A005148 G.f. = x + 47*x^2 + 2488*x^3 + 138799*x^4 + 7976456*x^5 + 467232200*x^6 + ...
%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}]
%t A005148 a[ n_] := If[ n < 1, 0, SeriesCoefficient[ ComposeSeries[ Series[ ((Pi / (2 EllipticK[m]))^2 / (1 - 2 m) - 1) / 24, {m, 0, n}], InverseSeries[ Series[ (1 - m) m/16, {m, 0, n}]]], {m, 0, n}]]; (* _Michael Somos_, Jul 06 2014 *)
%t A005148 a[ n_] := If[ n < 1, 0, SeriesCoefficient[ ComposeSeries[ Series[ ((Pi / (2 EllipticK[m]))^2 / (1 + m) - 1) / 24, {m, 0, n}], InverseSeries[ Series[ -(1 - m)^-2 m/16, {m, 0, n}]]], {m, 0, n}]]; (* _Michael Somos_, Jul 06 2014 *)
%o A005148 (PARI) {a(n) = if( n<1, 0, 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))} /* Broadhurst 2002 */
%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; )}; /* _Michael Somos_, Nov 05 2002 */
%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])}; /* _Michael Somos_, Mar 09 2004 */
%Y A005148 Cf. A005149, A076657, A018900.
%Y A005148 CF. A060236 (reduced mod 3).
%K A005148 nonn,easy,nice
%O A005148 0,3
%A A005148 _Simon Plouffe_ and _N. J. A. Sloane_
%E A005148 More terms from _Michael Somos_, Nov 24 2001