A377826 - OEIS

%I #11 Feb 16 2025 08:34:07

%S 1,2,7,49,489,6521,108643,2178107,51084337,1373054833,41624314371,

%T 1405311853595,52299954524953,2127347522554073,93902399411048803,

%U 4470613587492385051,228362858274694209249,12458393118650371672673,722983769486947261178371

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

%F E.g.f.: -LambertW(-x*(1+x))/x.

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

%F a(n) ~ sqrt(-2*sqrt(1 + 4*exp(-1)) + 2 + 8*exp(-1)) * 2^n * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(n+1) * exp(n - 1/2)). - _Vaclav Kotesovec_, Nov 09 2024

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

%Y Cf. A377827, A377828.

%Y Cf. A352410, A362771.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 09 2024