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%I #12 Nov 28 2024 11:05:35
%S 1,2,5,15,45,141,457,1520,5159,17797,62218,219946,784890,2823666,
%T 10229733,37289210,136665195,503301525,1861550310,6912114330,
%U 25755891510,96279420870,360961760250,1356920443944,5113523451462,19314197605826,73105593170132,277253358421060
%N Expansion of (1+x^3*C)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%F For n > 0, a(n) = 3*binomial(2*n-4,n-3)/n+2*binomial(2*n+1,n)/(n+2). - _Tani Akinari_, Nov 28 2024
%p a:= proc(n) option remember; `if`(n<3, 1+n^2,
%p 2*(2*n-5)*(67*n^3-93*n^2-28*n+12)*a(n-1)/
%p ((n+2)*(67*n^3-294*n^2+359*n-120)))
%p end:
%p seq(a(n), n=0..27); # _Alois P. Heinz_, Nov 28 2024
%o (Maxima) a(n):=if n=0 then 1 else 3*binomial(2*n-4,n-3)/n+2*binomial(2*n+1,n)/(n+2);
%o makelist(a(n),n,0,50); /* _Tani Akinari_, Nov 28 2024 */
%Y Cf. A000108.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2002