A143583 Apéry-like numbers: a(n) = (1/C(2n,n))*Sum_{k=0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).

1, 12, 164, 2352, 34596, 516912, 7806224, 118803648, 1818757924, 27972399792, 431824158864, 6686855325888, 103814819552016, 1615296581684928, 25180747436810304, 393189646497706752, 6148451986328464164, 96269310864931432368, 1509065592479205772304

Offset: 0

Author

Peter Bala, Aug 25 2008

Keywords

• easy
• nonn

Comments

These numbers bear some analogy to the Apéry numbers A005258. They appear in the evaluation of the spectral zeta function of the non-commutative harmonic oscillator zeta_Q(s) at s = 2 and satisfy a recurrence relation similar to the one satisfied by the Apéry numbers.

Examples

			G.f. = 1 + 12*x + 164*x^2 + 2352*x^3 + 34596*x^4 + 516912*x^5 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• K. Kimoto and M. Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, arXiv:math/0603700 [math.NT], 2006; Kyushu J. Math. Vol. 60, 2006, 383-404.
• Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Gn.
• Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.
• Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Gn.
• Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020.

Crossrefs

Cf. A005258.

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.)

Programs

Maple

a := n -> 1/binomial(2*n, n)*add(binomial(2*k, k)*binomial(4*k, 2*k)*binomial(2*n-2*k, n-k)*binomial(4*n-4*k, 2*n-2*k), k = 0..n): seq(a(n), n = 0..25);
series( 2*EllipticK(4*x^(1/2))/(Pi*sqrt(1-16*x)), x=0, 20); # Mark van Hoeij, Apr 06 2013
A143583 := n -> 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1):
seq(simplify(A143583(n)), n = 0..18); # Peter Luschny, Nov 12 2022

Mathematica

Table[1/Binomial[2*n,n]*Sum[Binomial[2*k,k]*Binomial[4*k,2*k]*Binomial[2*n-2*k,n-k]*Binomial[4*n-4*k,2*n-2*k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2013 *)

Formula

a(n) = (1/C(2n,n))*sum {k = 0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).

Recurrence relation:

a(0) = 1, a(1) = 12, n^2*a(n) = 4*(8*n^2-8*n+3)*a(n-1) - 256*(n-1)^2*a(n-2).

Congruences:

For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N.

a(n) ~ 16^n/(Pi*sqrt(Pi*n)) * (log(n) + gamma + 6*log(2)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 11 2013

a(n) = sum {k = 0..n} 4^(n-k) C(2k,k)^2*C(2n-2k,n-k). - Tito Piezas III, Dec 12 2014

a(n) = hypergeom([1/2,1/2,n+1],[1,n+3/2],1)*2^(5*n+1)*n!/((2*n+1)!!*Pi) - G. A. Edgar, Dec 10 2016

a(n) = binomial(4*n,2*n)*hypergeom([1/4,3/4,-n,-n], [1,1/4-n,3/4-n], 1). - Peter Luschny, May 14 2020

From Peter Luschny, Nov 12 2022: (Start)

a(n) = 16^n*Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2*binomial(n, k).

a(n) = 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1). (End)