A380078 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x))^(1/3) ).
1, 1, 8, 115, 2484, 72005, 2626846, 115688349, 5974568552, 354154378249, 23704428986010, 1768459611322481, 145525743200753356, 13095070459815108141, 1279226572751177845718, 134827003107939467441845, 15250595677663579282034256, 1842758049329907303778372625
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-3*x*exp(x))^(1/3))/x)) -
PARI
a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n/3+k+1/3, k)/((n+3*k+1)*(n-k)!));
Formula
E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) )^(1/3).
a(n) = n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n/3+k+1/3,k)/( (n+3*k+1)*(n-k)! ).
a(n) = (n!/(n+1)) * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-n/3-1/3,k)/(n-k)!.