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A260832 a(n) = numerator(Jtilde2(n)).

%I A260832 #51 Dec 08 2022 19:12:15
%S A260832 1,3,41,147,8649,32307,487889,1856307,454689481,1748274987,
%T A260832 26989009929,104482114467,6488426222001,25239009088827,
%U A260832 393449178700161,1535897056631667,1537112996582116041,6016831929058214523,94316599529950360769,369994845516850143483,23244865440911268112681
%N A260832 a(n) = numerator(Jtilde2(n)).
%C A260832 Jtilde2(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(2), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.
%H A260832 G. C. Greubel, <a href="/A260832/b260832.txt">Table of n, a(n) for n = 0..830</a>
%H A260832 Takashi Ichinose and 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 A260832 Kazufumi Kimoto and 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 1).
%H A260832 Ling Long, Robert Osburn and Holly Swisher, <a href="https://doi.org/10.1090/proc/13198">On a conjecture of Kimoto and Wakayama</a>, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.
%F A260832 Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
%F A260832 And 4n^2*J2(n) - (8n^2-8n+3)*J2(n-1) + 4(n-1)^2*J2(n-2) = 0 with J2(0) = 3*zeta(2) and J2(1) = 9*zeta(2)/4.
%F A260832 Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
%F A260832 Jtilde2(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(4*k,2*k)*binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))/(2^(4*n)*binomial(2*n,n)).
%F A260832 From _Andrey Zabolotskiy_, Oct 04 2016 and Dec 08 2022: (Start)
%F A260832 Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
%F A260832 G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
%F A260832 Jtilde2(n) = A143583(n) / 16^n. (End)
%F A260832 a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - _Peter Luschny_, Dec 08 2022
%p A260832 a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
%p A260832 seq(a(n), n = 0..20); # _Peter Luschny_, Dec 08 2022
%t A260832 Numerator[Table[Sum[ (-1)^k*Binomial[-1/2, k]^2*Binomial[n, k], {k, 0, n}], {n,0,50}]] (* _G. C. Greubel_, Feb 15 2017 *)
%o A260832 (PARI) a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2,k)^2*binomial(n, k)));
%o A260832 (PARI) a(n) = numerator(sum(k=0, n, binomial(2*k, k)*binomial(4*k, 2*k)* binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))) / (2^(4*n)* binomial(2*n,n)));
%Y A260832 Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).
%Y A260832 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 A260832 nonn,frac
%O A260832 0,2
%A A260832 _Michel Marcus_, Nov 17 2015