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

0,3

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.

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] // Denominator, {n, 0, 19}] (* Jean-François Alcover, Jul 16 2013, from the non-recursive formula *)

Crossrefs

Cf. A059415, A005259.

Sequence in context: A316297 A180170 A277174 * A240889 A108019 A241104

Adjacent sequences: A059413 A059414 A059415 * A059417 A059418 A059419

Keyword

nonn,frac

Author

N. J. A. Sloane, Jan 30 2001

Status

approved