cp's OEIS Frontend

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).

%I A143583 #62 Jan 29 2024 20:45:26
%S A143583 1,12,164,2352,34596,516912,7806224,118803648,1818757924,27972399792,
%T A143583 431824158864,6686855325888,103814819552016,1615296581684928,
%U A143583 25180747436810304,393189646497706752,6148451986328464164,96269310864931432368,1509065592479205772304
%N 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).
%C A143583 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.
%H A143583 Vincenzo Librandi, <a href="/A143583/b143583.txt">Table of n, a(n) for n = 0..200</a>
%H A143583 K. Kimoto and M. Wakayama, <a href="https://arxiv.org/abs/math/0603700">Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators</a>, arXiv:math/0603700 [math.NT], 2006; Kyushu J. Math. Vol. 60, 2006, 383-404.
%H A143583 Ji-Cai Liu and He-Xia Ni, <a href="https://arxiv.org/abs/2004.07652">Supercongruences for Almkvist--Zudilin sequences</a>, arXiv:2004.07652 [math.NT], 2020. See Gn.
%H A143583 Stéphane Ouvry and Alexios Polychronakos, <a href="https://arxiv.org/abs/2006.06445">Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers</a>, arXiv:2006.06445 [math-ph], 2020.
%H A143583 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020. See Gn.
%H A143583 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2005.02081">Congruences for two types of Apery-like sequences</a>, arXiv:2005.02081 [math.NT], 2020.
%F A143583 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).
%F A143583 Recurrence relation:
%F A143583 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).
%F A143583 Congruences:
%F A143583 For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N.
%F A143583 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
%F A143583 a(n) = sum {k = 0..n} 4^(n-k) C(2k,k)^2*C(2n-2k,n-k). - _Tito Piezas III_, Dec 12 2014
%F A143583 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
%F A143583 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
%F A143583 From _Peter Luschny_, Nov 12 2022: (Start)
%F A143583 a(n) = 16^n*Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2*binomial(n, k).
%F A143583 a(n) = 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1). (End)
%e A143583 G.f. = 1 + 12*x + 164*x^2 + 2352*x^3 + 34596*x^4 + 516912*x^5 + ...
%p A143583 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);
%p A143583 series( 2*EllipticK(4*x^(1/2))/(Pi*sqrt(1-16*x)), x=0, 20); # _Mark van Hoeij_, Apr 06 2013
%p A143583 A143583 := n -> 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1):
%p A143583 seq(simplify(A143583(n)), n = 0..18); # _Peter Luschny_, Nov 12 2022
%t A143583 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 *)
%Y A143583 Cf. A005258.
%Y A143583 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 A143583 easy,nonn
%O A143583 0,2
%A A143583 _Peter Bala_, Aug 25 2008