%I A143583
%S A143583 1,12,164,2352,34596,516912,7806224,118803648,1818757924,27972399792,
%T A143583 431824158864,6686855325888,103814819552016,1615296581684928,
%U A143583 25180747436810304,393189646497706752,6148451986328464164
%N A143583 Apery-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 Apery 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 Apery numbers.
%H A143583 K. Kimoto and M. Wakayama, <a href="http://arxiv.org/abs/math.NT/0603700">
               Apery-like numbers arising from special values of spectral zeta functions 
               for non-commutative harmonic oscillators</a> Kyushu J. Math. Vol. 
               60, 2006, 383-404.
%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). 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.
%p A143583 a := n -> 1/binomial(2n,n)*add(binomial(2k,k)*binomial(4k,2k)*binomial(2n-2k,
               n-k)*binomial(4n-4k,2n-2k),k = 0..n): seq(a(n),n = 0..20):
%Y A143583 Cf. A005258.
%Y A143583 Sequence in context: A138455 A024221 A093152 this_sequence A046174 A055760 
               A056591
%Y A143583 Adjacent sequences: A143580 A143581 A143582 this_sequence A143584 A143585 
               A143586
%K A143583 easy,nonn
%O A143583 0,2
%A A143583 Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008