A381882 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * C(x)) ), where C(x) is the g.f. of A000108.
1, 4, 24, 175, 1428, 12525, 115468, 1103777, 10844715, 108860766, 1111722956, 11514401451, 120666441067, 1277161022725, 13633269293868, 146606818816257, 1586739194404521, 17271207134469417, 188942438655850740, 2076317084779878706, 22909617070555385010
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3*(1-sqrt(1-4*x))/(2*x)))/x)
Formula
G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * C(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+3,n-k)/(n+2*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/2, 1+n/2, -n], [2 + n, 4 + 2*n], -4)/(1 + n). - Stefano Spezia, Mar 09 2025