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

%I A059416 #15 Dec 04 2016 04:46:35
%S A059416 1,1,4,36,288,36000,800,1372000,2195200,2667168000,2667168000,
%T A059416 28400004864,3550000608000,311974053431040,7799351335776000,
%U A059416 7799351335776000,1134451103385600,306545704901339904000,6812126775585331200,233621887768698933504000
%N A059416 Denominators of sequence arising from Apery's proof that zeta(3) is irrational.
%D A059416 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.
%H A059416 Seiichi Manyama, <a href="/A059416/b059416.txt">Table of n, a(n) for n = 0..768</a>
%H A059416 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
%F A059416 (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1.
%e A059416 0, 6, 351/4, 62531/36, ...
%p A059416 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;
%t A059416 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 *)
%Y A059416 Cf. A059415, A005259.
%K A059416 nonn,frac
%O A059416 0,3
%A A059416 _N. J. A. Sloane_, Jan 30 2001