A196795 a(n) = Sum_{k=0..n} binomial(n,k)*3^k*(k+1)^(n-k).

1, 4, 22, 145, 1096, 9259, 85924, 865183, 9364864, 108173827, 1325589676, 17149360111, 233271228880, 3324545097475, 49493784653644, 767665750130839, 12376226335249024, 206967901014192643, 3583561993192959436, 64136093489935863583, 1184711492540805987856

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

O.g.f.: Sum_{n>=0} 3^n*x^n/(1 - (n+1)*x)^(n+1).

E.g.f.: exp(x + 3*x*exp(x)).

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

Mathematica

Table[Sum[Binomial[n, k]3^k (k+1)^(n-k), {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 12 2012 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*3^k*(k+1)^(n-k))}
(PARI) {a(n)=polcoeff(sum(m=0, n, 3^m*x^m/(1-(m+1)*x+x*O(x^n))^(m+1)), n)}
(PARI) {a(n)=n!*polcoeff(exp(x+3*x*exp(x+x*O(x^n))), n)}

Crossrefs

Cf. A080108, A196794, A245834.

Sequence in context: A081002 A222012 A057834 * A395889 A278396 A369617

Adjacent sequences: A196792 A196793 A196794 * A196796 A196797 A196798

Keyword

nonn

Author

Paul D. Hanna, Oct 06 2011

Status

approved