A005148 Sequence of coefficients arising in connection with a rapidly converging series for Pi.
0, 1, 47, 2488, 138799, 7976456, 467232200, 27736348480, 1662803271215, 100442427373480, 6103747246289272, 372725876150863808, 22852464771010647496, 1405886026610765892544, 86741060172969340021952
Offset: 0
Examples
G.f. = x + 47*x^2 + 2488*x^3 + 138799*x^4 + 7976456*x^5 + 467232200*x^6 + ...
References
- F. Beukers, Letter to D. Shanks, Mar 13 1984
- J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195; see Exercise 6(a).
- D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, p. 255-7,276
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..555 (first 101 terms from T. D. Noe)
- Barry Brent, Folder : "current draft"
- Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
- Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
- M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217.
- D. Shanks, Letter to N. J. A. Sloane, date unknown. Also includes some notes from N. J. A. Sloane.
- Index entries for sequences related to the number Pi
Programs
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Mathematica
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}] 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 *) 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 *) -
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)}; -
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 */ -
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 */ -
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 */
Formula
a(n) = (1/24) * coefficient of x^n in Product_{k>=1} (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 = 0.078300067..., 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
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
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
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
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
Extensions
More terms from Michael Somos, Nov 24 2001
Comments