A378019 - OEIS

%I #10 Feb 16 2025 08:34:07

%S 1,2,9,79,957,15441,309943,7468301,210221385,6774449185,246049105131,

%T 9947338595085,443121311695021,21568178966624993,1138938283455953919,

%U 64856665518838006861,3961941908569940501649,258453847468153873181889,17932482767578645884498643

%N E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x) * A(x) ).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: (1+x) * exp( -LambertW(-x * (1+x)^2) ).

%F a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k+1,n-k)/k!.

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)*exp(-lambertw(-x*(1+x)^2))))

%o (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(2*k+1, n-k)/k!);

%Y Cf. A362772, A377827.

%Y Cf. A377963.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 14 2024