A240921 G.f.: Sum_{n>=0} n^n * x^n * (1 + n*x)^n / (1 + n*x + n^2*x^2)^(n+1).

1, 1, 3, 18, 146, 1530, 19620, 297360, 5201784, 103146120, 2286181800, 56011087440, 1503057473280, 43844234353920, 1381310964633600, 46743301840435200, 1690919874777893760, 65116170597269151360, 2659604669692822051200, 114838104572526535200000, 5226654647185285702752000

Offset

0,3

Comments

a(n) is divisible by [n/2]!.

Compare definition to the following identity, which holds for all fixed k:

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

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k,k).

a(n) ~ c * d^n * n^n / exp(n), where d = r*(-1+3*r-3*r^2)/(1-3*r+2*r^2) = 2.334305682349197638435662..., where r = 0.722795640379451585372396... is the root of the equation (1-r) * (r + 1/LambertW(-exp(-1/r)/r)) + (2*r-1)^2 = 0, and c = 1.04764685950245700560418116727397... . - Vaclav Kotesovec, Aug 03 2014

Example

G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 146*x^4 + 1530*x^5 + 19620*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2)^2 + 2^2*x^2*(1+2*x)^2/(1+2*x+4*x^2)^3 + 3^3*x^3*(1+3*x)^3/(1+3*x+9*x^2)^4 + 4^4*x^4*(1+4*x)^4/(1+4*x+16*x^2)^5 +...

Mathematica

Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 03 2014 *)

Prog

(PARI) /* From definition: */
{a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(1+m*x)^m/(1 + m*x + m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A240172, A240956, A240957, A240958, A240921, A008277 (Stirling2).

Sequence in context: A074525 A118331 A371825 * A221961 A138420 A152409

Adjacent sequences: A240918 A240919 A240920 * A240922 A240923 A240924

Keyword

nonn

Author

Paul D. Hanna, Aug 02 2014

Status

approved