|
%I #62 Jan 29 2024 20:45:26
%S 1,12,164,2352,34596,516912,7806224,118803648,1818757924,27972399792,
%T 431824158864,6686855325888,103814819552016,1615296581684928,
%U 25180747436810304,393189646497706752,6148451986328464164,96269310864931432368,1509065592479205772304
%N 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 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 Vincenzo Librandi, <a href="/A143583/b143583.txt">Table of n, a(n) for n = 0..200</a>
%H 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 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 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 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 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 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 Recurrence relation:
%F 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 Congruences:
%F For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N.
%F 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 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 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 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 From _Peter Luschny_, Nov 12 2022: (Start)
%F a(n) = 16^n*Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2*binomial(n, k).
%F a(n) = 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1). (End)
%e G.f. = 1 + 12*x + 164*x^2 + 2352*x^3 + 34596*x^4 + 516912*x^5 + ...
%p 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 series( 2*EllipticK(4*x^(1/2))/(Pi*sqrt(1-16*x)), x=0, 20); # _Mark van Hoeij_, Apr 06 2013
%p A143583 := n -> 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1):
%p seq(simplify(A143583(n)), n = 0..18); # _Peter Luschny_, Nov 12 2022
%t 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 Cf. A005258.
%Y 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 easy,nonn
%O 0,2
%A _Peter Bala_, Aug 25 2008
|
