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A264541 a(n) = numerator(Jtilde3(n)). 34
0, 1, 65, 13247, 704707, 660278641, 357852111131, 309349386395887, 240498440880062263, 148443546307725010253, 61760947097005048531, 13658972396318235617977, 723464275788899734058353751, 489812222050789870424202126629, 2614176630672654770175367214389, 204702102697072009862200307064701369 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..345

Takashi Ichinose, 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, 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 2).

FORMULA

Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).

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.

MATHEMATICA

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 *)

PROG

(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))));

CROSSREFS

Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264542 (denominators).

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: A251150 A176370 A093265 * A120801 A283580 A084225

Adjacent sequences:  A264538 A264539 A264540 * A264542 A264543 A264544

KEYWORD

nonn,frac

AUTHOR

Michel Marcus, Nov 17 2015

STATUS

approved

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Last modified October 20 19:44 EDT 2018. Contains 316401 sequences. (Running on oeis4.)