This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A264541 #18 Oct 24 2017 03:35:06
%S A264541 0,1,65,13247,704707,660278641,357852111131,309349386395887,
%T A264541 240498440880062263,148443546307725010253,61760947097005048531,
%U A264541 13658972396318235617977,723464275788899734058353751,489812222050789870424202126629,2614176630672654770175367214389,204702102697072009862200307064701369
%N A264541 a(n) = numerator(Jtilde3(n)).
%C A264541 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 A264541 G. C. Greubel, <a href="/A264541/b264541.txt">Table of n, a(n) for n = 0..345</a>
%H A264541 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 A264541 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 A264541 Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).
%F A264541 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 A264541 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 A264541 (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 A264541 Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264542 (denominators).
%Y A264541 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 A264541 nonn,frac
%O A264541 0,3
%A A264541 _Michel Marcus_, Nov 17 2015