login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 0
1, 1, 2, 9, 57, 421, 3593, 34557, 366832, 4251094, 53238166, 714702779, 10221402872, 154913725486, 2477047085038, 41629752595369, 732956458329580, 13480858878123068, 258362762534442843, 5148079352377053578, 106437899659055825010, 2279307634231962670724 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 24 05:38 EDT 2021. Contains 348217 sequences. (Running on oeis4.)