A381881 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * C(x)) ), where C(x) is the g.f. of A000108.
1, 3, 14, 82, 547, 3958, 30249, 240362, 1966235, 16449495, 140093989, 1210575512, 10587490383, 93540456103, 833619150838, 7484887130882, 67645312129491, 614872423359187, 5617522739173495, 51556112664387720, 475105557839611760, 4394434006611790855
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1-sqrt(1-4*x))/(2*x)))/x)
Formula
G.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * C(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(2*n+2,n-k)/(n+2*k+1).
a(n) = binomial(2*(1 + n), n)*hypergeom([(1+n)/2, 1+n/2, -n], [2 + n, 3 + n], -4)/(1 + n). - Stefano Spezia, Mar 09 2025