A389783 Expansion of e.g.f. exp( -LambertW(-x*(1+x)^2) ) / (1+x).

1, 0, 7, 37, 577, 8931, 185461, 4496745, 127887649, 4148705863, 151565099461, 6156548762013, 275357548277857, 13448461104526683, 712266298742636821, 40664499724608298801, 2489734005684242102977, 162742147349796184406799, 11311906857246605162342917

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Formula

E.g.f. A(x) satisfies A(x) = exp( x * (1+x)^3 * A(x) ) / (1+x).

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

a(n) ~ sqrt((1+r)*(1+3*r)) * n^(n-1) / (exp(n-2) * r^(n-1)), where r = 0.23946298617885055543944358... is the real root of the equation r*(1+r)^2 = exp(-1). - Vaclav Kotesovec, Oct 15 2025

Mathematica

Table[n!*Sum[(k+1)^(k-1)*Binomial[2*k-1, n-k]/k!, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Nov 01 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(2*k-1, n-k)/k!);
(Magma) [Factorial(n) * &+[(k+1)^(k-1)* Binomial(2*k-1, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Nov 01 2025

Crossrefs

Cf. A108919, A362682, A389782.

Sequence in context: A341377 A082687 A117731 * A155010 A292807 A210620

Adjacent sequences: A389780 A389781 A389782 * A389784 A389785 A389786

Keyword

nonn

Author

Seiichi Manyama, Oct 14 2025

Status

approved