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%I #18 Oct 24 2017 03:35:06
%S 0,1,65,13247,704707,660278641,357852111131,309349386395887,
%T 240498440880062263,148443546307725010253,61760947097005048531,
%U 13658972396318235617977,723464275788899734058353751,489812222050789870424202126629,2614176630672654770175367214389,204702102697072009862200307064701369
%N a(n) = numerator(Jtilde3(n)).
%C Jtilde3(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(3), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.
%H G. C. Greubel, <a href="/A264541/b264541.txt">Table of n, a(n) for n = 0..345</a>
%H Takashi Ichinose, Masato Wakayama, <a href="http://doi.org/10.2206/kyushujm.59.39">Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations</a>, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
%H Kazufumi Kimoto, Masato Wakayama, <a href="http://doi.org/10.2206/kyushujm.60.383">Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators</a>, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).
%F Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).
%F 4n^2*J3(n) - (8n^2-8n+3)*J3(n-1) + 4(n-1)^2*J3(n-2) = 2^n*(n-1)!/(2n-1)!! with J3(0)=7*zeta(3) and J3(1)=21*zeta(3)/4 + 1/2.
%t Numerator[Table[-2*Sum[(-1)^k*Binomial[-1/2, k]^2*Binomial[n, k]*Sum[ 1/(Binomial[-1/2, j]^2*(2*j + 1)^3), {j, 0, k - 1}], {k, 0, n}], {n, 0, 50}]] (* _G. C. Greubel_, Oct 24 2017 *)
%o (PARI) a(n) = numerator(-2*sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)*sum(j=0, k-1, 1/(binomial(-1/2,j)^2*(2*j+1)^3))));
%Y Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264542 (denominators).
%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%K nonn,frac
%O 0,3
%A _Michel Marcus_, Nov 17 2015
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