A222078 O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!.

1, 5, 47, 742, 16357, 459369, 15651260, 626935936, 28872594389, 1503262704775, 87328047029511, 5600639046765690, 393092088068927860, 29974039720132943036, 2467669218502361588472, 218168186315818183909344, 20617367868151866462395205, 2074120178028300166990286691

Offset

0,2

Links

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

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+4)^k * x^k / (1 + k*(k+4)*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+4)^n.

a(n) ~ n^n * 2^(2*n+3/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

Example

O.g.f.: A(x) = 1 + 5*x + 47*x^2 + 742*x^3 + 16357*x^4 + 459369*x^5 +...
where
A(x) = 1 + 5*x*exp(-5*x) + 12^2*exp(-12*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 32^4*exp(-32*x)*x^4/4! + 45^5*exp(-45*x)*x^5/5! +...
is a power series in x with integer coefficients.

Mathematica

Flatten[{1, Table[Sum[Binomial[n, j] * 4^(n-j) * StirlingS2[n+j, n], {j, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+4)^m*x^m*exp(-m*(m+4)*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+4)^k*x^k/(1+k*(k+4)*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+4)^n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A217903, A217900, A222076, A222077, A222079.

Sequence in context: A246040 A183773 A247982 * A061572 A140422 A145564

Adjacent sequences: A222075 A222076 A222077 * A222079 A222080 A222081

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2013

Status

approved