A059416 Denominators of sequence arising from Apery's proof that zeta(3) is irrational.

1, 1, 4, 36, 288, 36000, 800, 1372000, 2195200, 2667168000, 2667168000, 28400004864, 3550000608000, 311974053431040, 7799351335776000, 7799351335776000, 1134451103385600, 306545704901339904000, 6812126775585331200, 233621887768698933504000

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..768
• V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

Crossrefs

Cf. A059415, 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] // Denominator, {n, 0, 19}] (* 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.