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A005148 Sequence of coefficients arising in connection with a rapidly converging series for Pi.
(Formerly M5290)
10
0, 1, 47, 2488, 138799, 7976456, 467232200, 27736348480, 1662803271215, 100442427373480, 6103747246289272, 372725876150863808, 22852464771010647496, 1405886026610765892544, 86741060172969340021952 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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)

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

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)

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

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

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

EXAMPLE

G.f. = x + 47*x^2 + 2488*x^3 + 138799*x^4 + 7976456*x^5 + 467232200*x^6 + ...

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 *)

PROG

(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 */

CROSSREFS

Cf. A005149, A076657, A018900.

A060236 (reduced mod 3).

Sequence in context: A009991 A052463 A327770 * A123798 A104069 A334180

Adjacent sequences:  A005145 A005146 A005147 * A005149 A005150 A005151

KEYWORD

nonn,easy,nice

AUTHOR

Simon Plouffe and N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos, Nov 24 2001

Typo in numerical value of constant "a" corrected by Vaclav Kotesovec, Jul 28 2013

STATUS

approved

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Last modified July 3 16:27 EDT 2022. Contains 355055 sequences. (Running on oeis4.)