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

0,3

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)).

Links

Table of n, a(n) for n=0..21.

Example

 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 +...

Prog

 (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), ", "))

Crossrefs

Cf. A218670.

Sequence in context: A175896 A175900 A175912 * A111545 A070075 A124405

Adjacent sequences: A218821 A218822 A218823 * A218825 A218826 A218827

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2012

Status

approved