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

Author

N. J. A. Sloane, Jan 30 2001

Keywords

• nonn
• frac

Examples

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

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.

Crossrefs

Cf. A059416, A005259.

Programs

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 *)

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.