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 A260832 a(n) = numerator(Jtilde2(n)). 36
 1, 3, 41, 147, 8649, 32307, 487889, 1856307, 454689481, 1748274987, 26989009929, 104482114467, 6488426222001, 25239009088827, 393449178700161, 1535897056631667, 1537112996582116041, 6016831929058214523, 94316599529950360769, 369994845516850143483, 23244865440911268112681 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3). 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.) Sequence in context: A262555 A343814 A106978 * A089131 A057650 A280176 Adjacent sequences: A260829 A260830 A260831 * A260833 A260834 A260835 KEYWORD nonn,frac AUTHOR Michel Marcus, Nov 17 2015 STATUS approved

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Last modified July 24 08:15 EDT 2024. Contains 374575 sequences. (Running on oeis4.)