A195263 E.g.f.: Sum_{n>=0} 3^n*(exp(n*x) - 1)^n.

1, 3, 75, 4809, 578415, 112024353, 31851411375, 12493267169169, 6464106627329055, 4265281191267407073, 3495556570494504442575, 3483310917470882398369329, 4147647341931988462919773695, 5815857702618060221437908948993, 9485411994735540168549266106329775

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Formula

a(n) = Sum_{k=0..n} 3^k*k^n*k!*Stirling2(n,k).

a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.36805443792839553744923868309093616341812244322234916... - Vaclav Kotesovec, Oct 04 2020

Example

E.g.f.: A(x) = 1 + 3*x + 75*x^2/2! + 4809*x^3/3! + 578415*x^4/4! +...
where
A(x) = 1 + 3*(exp(x)-1) + 3^2*(exp(2*x)-1)^2 + 3^3*(exp(3*x)-1)^3 +...

Mathematica

Flatten[{1, Table[Sum[3^k * k^n * k! * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 04 2020 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 3^m*(exp(m*X)-1)^m), n)}
(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, 3^k*k^n*k!*Stirling2(n, k))}

Crossrefs

Cf. A195005, A122399, A301582, A338040.

Sequence in context: A228841 A361537 A249938 * A034940 A300924 A183290

Adjacent sequences: A195260 A195261 A195262 * A195264 A195265 A195266

Keyword

nonn

Author

Paul D. Hanna, Sep 13 2011

Status

approved