OFFSET
0,2
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..830
Takashi Ichinose and Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto and Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 1).
Ling Long, Robert Osburn and Holly Swisher, On a conjecture of Kimoto and Wakayama, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.
FORMULA
Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
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.
Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
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)).
From Andrey Zabolotskiy, Oct 04 2016 and Dec 08 2022: (Start)
Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
Jtilde2(n) = A143583(n) / 16^n. (End)
a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - Peter Luschny, Dec 08 2022
MAPLE
a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
seq(a(n), n = 0..20); # Peter Luschny, Dec 08 2022
MATHEMATICA
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 *)
PROG
(PARI) a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)));
(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)));
CROSSREFS
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.)
KEYWORD
nonn,frac
AUTHOR
Michel Marcus, Nov 17 2015
STATUS
approved