A360058 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+2,n-k) * Catalan(k).
1, 2, 2, 2, 3, 3, 2, 4, 5, 0, 4, 13, -7, -7, 48, -16, -93, 180, 74, -584, 517, 1111, -2850, 207, 8281, -10738, -11740, 46967, -22167, -115845, 211052, 94468, -766989, 660110, 1554938, -3983408, 121429, 12272689, -15692006, -18841086, 72792247, -31828764
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+2, n-k)*binomial(2*k, k)/(k+1));
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PARI
my(N=50, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1+4*x/(1-x)^3))))
Formula
a(n) = binomial(n+2,2) - Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^3 - x * A(x)^2.
G.f.: 2 / ( (1-x)^3 * (1 + sqrt( 1 + 4*x/(1-x)^3 )) ).
D-finite with recurrence (n+1)*a(n) -2*a(n-1) +2*(n-3)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jan 25 2023