This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380078 #12 Jan 12 2025 07:56:32
%S A380078 1,1,8,115,2484,72005,2626846,115688349,5974568552,354154378249,
%T A380078 23704428986010,1768459611322481,145525743200753356,
%U A380078 13095070459815108141,1279226572751177845718,134827003107939467441845,15250595677663579282034256,1842758049329907303778372625
%N A380078 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x))^(1/3) ).
%F A380078 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) )^(1/3).
%F A380078 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)! ).
%F A380078 a(n) = (n!/(n+1)) * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-n/3-1/3,k)/(n-k)!.
%o A380078 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-3*x*exp(x))^(1/3))/x))
%o A380078 (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)!));
%Y A380078 Cf. A213644, A380077.
%Y A380078 Cf. A380039.
%K A380078 nonn
%O A380078 0,3
%A A380078 _Seiichi Manyama_, Jan 11 2025