A222079 O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.

1, 6, 62, 1040, 24076, 703800, 24786512, 1020779520, 48130232528, 2557117300640, 151180506557280, 9846055638729216, 700523098562671360, 54066239308284456960, 4499576117943522662400, 401709919258066014720000, 38299206898825369235170560, 3883999501445283274005895680

Offset: 0

Author

Paul D. Hanna, Feb 24 2013

Keywords

• nonn

Comments

From Vaclav Kotesovec, May 22 2014: (Start)

Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+p)^n = Sum_{j=0..n} binomial(n,j) * p^(n-j) * StirlingS2(n+j,n).

a(n) ~ n^n * 2^(2*n+p/2) / (sqrt(2*Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...

(End)

Examples

			O.g.f.: A(x) = 1 + 6*x + 62*x^2 + 1040*x^3 + 24076*x^4 + 703800*x^5 +...
where
A(x) = 1 + 6*x*exp(-6*x) + 14^2*exp(-14*x)*x^2/2! + 24^3*exp(-24*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 50^5*exp(-50*x)*x^5/5! +...
is a power series in x with integer coefficients.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..340

Crossrefs

Cf. A217904, A217900, A222076, A222077, A222078.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[n,j] * 5^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

PARI

{a(n)=polcoeff(sum(m=0, n, m^m*(m+5)^m*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+5)^k*x^k/(1+k*(k+5)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+5)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+5)^k * x^k / (1 + k*(k+5)*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+5)^n.

a(n) ~ n^n * 2^(2*n+2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+5/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014