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A005148
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Sequence of coefficients arising in connection with a rapidly converging series for Pi.
(Formerly M5290)
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10
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0, 1, 47, 2488, 138799, 7976456, 467232200, 27736348480, 1662803271215, 100442427373480, 6103747246289272, 372725876150863808, 22852464771010647496, 1405886026610765892544, 86741060172969340021952
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OFFSET
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0,3
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COMMENTS
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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)
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REFERENCES
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F. Beukers, Letter to D. Shanks, Mar 13 1984
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
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Index entries for sequences related to the number Pi
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FORMULA
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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, 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, Oct 24 2002
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MATHEMATICA
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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}]
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PROG
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(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])
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CROSSREFS
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Cf. A005149.
A060236 (reduced mod 3).
Sequence in context: A049668 A009991 A052463 * A123798 A104069 A194617
Adjacent sequences: A005145 A005146 A005147 * A005149 A005150 A005151
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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Simon Plouffe and N. J. A. Sloane.
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EXTENSIONS
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More terms from Michael Somos, Nov 24 2001
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STATUS
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approved
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