A344504 a(n) = [x^n] ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)).

0, 1, 6, 26, 100, 361, 1254, 4245, 14108, 46247, 149998, 482412, 1540880, 4893859, 15468910, 48696930, 152764452, 477771447, 1490245302, 4637349186, 14400224496, 44632551567, 138101593398, 426658380621, 1316306945952, 4055853282741, 12482506508174, 38375733088400

Offset

0,3

Comments

Motzkin transform of the squares.

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = Sum_{k=0..n} k^2*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

a(n) ~ 4 * 3^(n - 1/2) * sqrt(n/Pi) * (1 - sqrt(3*Pi/n)/2). - Vaclav Kotesovec, May 24 2021

D-finite with recurrence -(n+1)*(2*n-3)*a(n) +(10*n^2-5*n-12)*a(n-1) -3*(2*n+5)*(n-1)*a(n-2) -9*(2*n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Mar 06 2022

Maple

gf := ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n=0..27);

Crossrefs

Cf. A064189 (Motzkin numbers).

Keyword

nonn

Author

Peter Luschny, May 23 2021

Status

approved