|
%I #8 Nov 02 2021 07:33:19
%S 1,0,-1,3,0,-5,15,0,-28,84,0,-165,495,0,-1001,3003,0,-6188,18564,0,
%T -38760,116280,0,-245157,735471,0,-1562275,4686825,0,-10015005,
%U 30045015,0,-64512240,193536720,0,-417225900,1251677700,0,-2707475148,8122425444,0,-17620076360
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
%C Inverse binomial convolution of the Motzkin numbers.
%F a(3*n) = binomial(3*n, n) (A005809).
%F a(3*n - 1) = -binomial(3*n - 1, n - 1) (A025174).
%F a(3*n - 2) = 0.
%F Conjecture D-finite with recurrence -18*(2*n+1) *(2*n-1) *(n+1) *a(n) +2*(-36*n^3+554*n^2-1128*n+27) *a(n-1) +6*(-12*n^3-188*n^2+1235*n-1618) *a(n-2) +9*(54*n^3-27*n^2-183*n+320) *a(n-3) +54*(n-3) *(9*n^2-125*n+75) *a(n-4) +81 *(n-3) *(n-4) *(6*n+127) *a(n-5)=0. - _R. J. Mathar_, Nov 02 2021
%p a := n -> add((-1)^(n - k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n): seq(simplify(a(n)), n = 0..41);
%Y Cf. A064189 (Motzkin numbers), A005809, A025174, A344502.
%K sign
%O 0,4
%A _Peter Luschny_, May 23 2021
|
