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A264542 a(n) = denominator(Jtilde3(n)).

%I A264542 #21 Oct 24 2017 08:38:58
%S A264542 1,2,96,17280,860160,774144000,408748032000,347163328512000,
%T A264542 266621436297216000,163172319013896192000,67488959156767948800,
%U A264542 14865958099336613068800,785345441564243189248819200,530893518497428395932201779200,2831432098652951444971742822400,221701133324526098141287462993920000
%N A264542 a(n) = denominator(Jtilde3(n)).
%C A264542 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 A264542 G. C. Greubel, <a href="/A264542/b264542.txt">Table of n, a(n) for n = 0..345</a>
%H A264542 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 A264542 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 A264542 Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).
%F A264542 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 A264542 Denominator[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 23 2017 *)
%o A264542 (PARI) a(n) = denominator(-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 A264542 Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264541 (numerators).
%Y A264542 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 A264542 nonn,frac
%O A264542 0,2
%A A264542 _Michel Marcus_, Nov 17 2015