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A059416 Denominators of sequence arising from Apery's proof that zeta(3) is irrational. 8
1, 1, 4, 36, 288, 36000, 800, 1372000, 2195200, 2667168000, 2667168000, 28400004864, 3550000608000, 311974053431040, 7799351335776000, 7799351335776000, 1134451103385600, 306545704901339904000, 6812126775585331200, 233621887768698933504000 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 21 10:33 EDT 2018. Contains 316414 sequences. (Running on oeis4.)