A196794 a(n) = Sum_{k=0..n} binomial(n,k)*2^k*(k+1)^(n-k).

1, 3, 13, 69, 425, 2953, 22701, 190445, 1725777, 16757649, 173244629, 1896821941, 21897166137, 265525063001, 3371067773565, 44683137692157, 616811052816545, 8847765111928609, 131622808197394341, 2027097866771329349, 32267707989783480201, 530125689222591861993

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..523

Formula

O.g.f.: Sum_{n>=0} 2^n*x^n/(1 - (n+1)*x)^(n+1).

E.g.f.: exp(x + 2*x*exp(x)).

(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2^(3/2)))) / (2*LambertW(sqrt(n)/2^(3/2))). - Vaclav Kotesovec, Oct 16 2025

Maple

S:= series(exp(x+2*x*exp(x)), x, 51):
seq(coeff(S, x, j)*j!, j=0..50); # Robert Israel, Jan 20 2017

Mathematica

Table[Sum[Binomial[n, k]*2^k*(k+1)^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2025 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*2^k*(k+1)^(n-k))}
(PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*x^m/(1-(m+1)*x+x*O(x^n))^(m+1)), n)}
(PARI) {a(n)=n!*polcoeff(exp(x+2*x*exp(x+x*O(x^n))), n)}

Crossrefs

Cf. A080108, A196795, A240165.

Sequence in context: A067145 A192739 A088368 * A184818 A352370 A007808

Adjacent sequences: A196791 A196792 A196793 * A196795 A196796 A196797

Keyword

nonn

Author

Paul D. Hanna, Oct 06 2011

Status

approved