A217902 O.g.f.: Sum_{n>=0} 3*n^n*(n+3)^(n-1) * exp(-n*(n+3)*x) * x^n / n!.

1, 3, 18, 210, 3696, 86436, 2521800, 88274640, 3608360064, 168822613872, 8901871248480, 522534101560224, 33804242536287744, 2390169742849449216, 183412961210465667072, 15183107016739655860224, 1348837954231568133427200, 128012762381954718934183680

Offset

0,2

Comments

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 3*(n+3)^(n-1) * exp(-(n+3)*x) * x^n/n!.

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

Formula

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

a(n) = 1/n! * [x^n] Sum_{k>=0} 3*k^k*(k+3)^(k-1)*x^k / (1 + k*(k+3)*x)^(k+1).

a(n) = [x^n] 1 + 3*x*(1+3*x)^(n-1) / Product_{k=1..n} (1-k*x).

a(n) = [x^n] 1 + 3*x*(1-3*x)^(n-1) / Product_{k=1..n} (1-(k+3)*x).

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

Example

O.g.f.: A(x) = 1 + 3*x + 18*x^2 + 210*x^3 + 3696*x^4 + 86436*x^5 + 2521800*x^6 +...
where
A(x) = 1 + 3*1^1*4^0*x*exp(-1*4*x) + 3*2^2*5^1*exp(-2*5*x)*x^2/2! + 3*3^3*6^2*exp(-3*6*x)*x^3/3! + 3*4^4*7^3*exp(-4*7*x)*x^4/4! + 3*5^5*8^4*exp(-5*8*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.

Mathematica

Flatten[{1, Table[Sum[Binomial[n-1, j]*3^(n-j)*StirlingS2[n+j, n], {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, 3*m^m*(m+3)^(m-1)*x^m*exp(-m*(m+3)*x+x*O(x^n))/m!), n)}
(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, 3*k^k*(k+3)^(k-1)*x^k/(1+k*(k+3)*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=1/n!*sum(k=0, n, 3*(-1)^(n-k)*binomial(n, k)*k^n*(k+3)^(n-1))}
(PARI) {a(n)=polcoeff(1+3*x*(1+3*x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(1+3*x*(1-3*x)^n/prod(k=0, n, 1-(k+3)*x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A217899, A217900, A217901, A217903, A217904, A217905, A217910.

Sequence in context: A135077 A157538 A024488 * A356614 A183241 A163883

Adjacent sequences: A217899 A217900 A217901 * A217903 A217904 A217905

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2012

Status

approved