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%I #10 Jan 11 2024 10:55:53
%S 0,1,6,26,100,361,1254,4245,14108,46247,149998,482412,1540880,4893859,
%T 15468910,48696930,152764452,477771447,1490245302,4637349186,
%U 14400224496,44632551567,138101593398,426658380621,1316306945952,4055853282741,12482506508174,38375733088400
%N a(n) = [x^n] ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)).
%C Motzkin transform of the squares.
%F a(n) = Sum_{k=0..n} k^2*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
%F a(n) ~ 4 * 3^(n - 1/2) * sqrt(n/Pi) * (1 - sqrt(3*Pi/n)/2). - _Vaclav Kotesovec_, May 24 2021
%F 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
%p gf := ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)):
%p ser := series(gf, x, 30): seq(coeff(ser, x, n), n=0..27);
%Y Cf. A064189 (Motzkin numbers).
%K nonn
%O 0,3
%A _Peter Luschny_, May 23 2021
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