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 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

%I

%S 1,1,2,9,57,421,3593,34557,366832,4251094,53238166,714702779,

%T 10221402872,154913725486,2477047085038,41629752595369,

%U 732956458329580,13480858878123068,258362762534442843,5148079352377053578,106437899659055825010,2279307634231962670724

%N 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).

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

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

%e O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 57*x^4 + 421*x^5 + 3593*x^6 +...

%e such that

%e 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! +...

%e where the partition function P(x) = Product_{n>=1} 1/(1-x^n) begins:

%e 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 +...

%o (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)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A218670.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 06 2012

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Last modified December 5 10:11 EST 2021. Contains 349543 sequences. (Running on oeis4.)