A222076 O.g.f.: Sum_{n>=0} n^n*(n+2)^n * exp(-n*(n+2)*x) * x^n / n!.

1, 3, 23, 320, 6397, 166467, 5338412, 203578776, 9001795829, 452924585465, 25555585227999, 1598279794889076, 109748572718377660, 8209004345714098500, 664396187060996529528, 57853075421585981420208, 5393119810256349152565573, 535908449308064099732283429, 56548822143306498413322880709

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..345

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+2)^k * x^k / (1 + k*(k+2)*x)^(k+1).

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

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

Example

O.g.f.: A(x) = 1 + 3*x + 23*x^2 + 320*x^3 + 6397*x^4 + 166467*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 8^2*exp(-8*x)*x^2/2! + 15^3*exp(-15*x)*x^3/3! + 24^4*exp(-24*x)*x^4/4! + 35^5*exp(-35*x)*x^5/5! +...
is a power series in x with integer coefficients.

Mathematica

Flatten[{1, Table[Sum[Binomial[n, j] * 2^(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+2)^m*x^m*exp(-m*(m+2)*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+2)^k*x^k/(1+k*(k+2)*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+2)^n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A217901, A217900, A222077, A222078, A222079.

Sequence in context: A088692 A188313 A227821 * A338301 A129458 A118184

Adjacent sequences: A222073 A222074 A222075 * A222077 A222078 A222079

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2013

Status

approved