A218824 O.g.f.: A(x) = Sum_{n>=0} n^n * x^n/n! * P(n*x)^n * exp(-n*x*P(n*x)), where P(x) is the partition function (A000041).

1, 1, 2, 9, 57, 421, 3593, 34557, 366832, 4251094, 53238166, 714702779, 10221402872, 154913725486, 2477047085038, 41629752595369, 732956458329580, 13480858878123068, 258362762534442843, 5148079352377053578, 106437899659055825010, 2279307634231962670724

Offset: 0

Author

Paul D. Hanna, Nov 06 2012

Keywords

• nonn

Comments

Compare the o.g.f. to the LambertW identity:

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

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 57*x^4 + 421*x^5 + 3593*x^6 +...
such that
A(x) = 1 + x*P(x)*exp(-x*P(x)) + 2^2*x^2*P(2*x)^2*exp(-2*x*P(2*x))/2! + 3^3*x^3*P(3*x)^3*exp(-3*x*P(3*x))/3! + 4^4*x^4*P(4*x)^4*exp(-4*x*P(4*x))/4! + 5^5*x^5*P(5*x)^5*exp(-5*x*P(5*x))/5! +...
where the partition function P(x) = Product_{n>=1} 1/(1-x^n) begins:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 +...
		

Crossrefs

Cf. A218670.

Programs

PARI

{a(n)=local(A=1+x);A=sum(k=0,n,k^k/eta(k*x+x*O(x^n))^k*x^k/k!*exp(-k*x/eta(k*x+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))