A375897 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/2)))^2.
1, 2, 12, 122, 1780, 34082, 810740, 23093562, 767175972, 29140904402, 1246366394548, 59292772664666, 3106206974812292, 177715679350850370, 11026719500616041076, 737552919428497318394, 52907911316906095281508, 4051998061642112552244722
Offset: 0
Programs
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Mathematica
Table[2/(n+2)! * Sum[(n + k + 1)!*StirlingS2[n, k], {k, 0, n} ], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^2)) -
PARI
a(n) = 2*sum(k=0, n, (n+k+1)!*stirling(n, k, 2))/(n+2)!;
Formula
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^2.
a(n) = (2/(n+2)!) * Sum_{k=0..n} (n+k+1)! * Stirling2(n,k).
a(n) ~ LambertW(2*exp(1))^(n+2) * n^(n-1) / (2^(n+1) * exp(n) * sqrt(LambertW(2*exp(1)) + 1) * (LambertW(2*exp(1)) - 1)^(2*n+2)). - Vaclav Kotesovec, Aug 27 2025