A059415 Numerators of sequence arising from Apery's proof that zeta(3) is irrational.

0, 6, 351, 62531, 11424695, 35441662103, 20637706271, 963652602684713, 43190915887542721, 1502663969043851254939, 43786938951280269198311, 13780864457900933987428453, 51520703555193710949642777493

Offset

0,2

References

M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..358

V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

Formula

(n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1.

Example

0, 6, 351/4, 62531/36, ...

Maple

a := proc(n) option remember; if n=0 then 0 elif n=1 then 6 else (n^(-3))* ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); fi; end;

Mathematica

a[n_] := Sum[ Binomial[n, k]^2*Binomial[k + n, k]^2*(Sum[1/m^3, {m, 1, n}] + Sum[(-1)^(m - 1)/(2*m^3*Binomial[n, m]*Binomial[m + n, m]), {m, 1, k}]), {k, 0, n}]; Table[a[n] // Numerator, {n, 0, 12}] (* Jean-François Alcover, Jul 16 2013, from the non-recursive formula *)

Crossrefs

Cf. A059416, A005259.

Sequence in context: A221923 A000409 A214445 * A246112 A197780 A197611

Adjacent sequences: A059412 A059413 A059414 * A059416 A059417 A059418

Keyword

nonn,frac

Author

N. J. A. Sloane, Jan 30 2001

Status

approved