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%I #51 Dec 08 2022 19:12:15
%S 1,3,41,147,8649,32307,487889,1856307,454689481,1748274987,
%T 26989009929,104482114467,6488426222001,25239009088827,
%U 393449178700161,1535897056631667,1537112996582116041,6016831929058214523,94316599529950360769,369994845516850143483,23244865440911268112681
%N a(n) = numerator(Jtilde2(n)).
%C 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 G. C. Greubel, <a href="/A260832/b260832.txt">Table of n, a(n) for n = 0..830</a>
%H 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 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 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 Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
%F 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 Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
%F 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 From _Andrey Zabolotskiy_, Oct 04 2016 and Dec 08 2022: (Start)
%F 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 G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
%F Jtilde2(n) = A143583(n) / 16^n. (End)
%F a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - _Peter Luschny_, Dec 08 2022
%p a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
%p seq(a(n), n = 0..20); # _Peter Luschny_, Dec 08 2022
%t 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 (PARI) a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2,k)^2*binomial(n, k)));
%o (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 Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).
%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 0,2
%A _Michel Marcus_, Nov 17 2015
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