cp's OEIS Frontend

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).

%I A218824 #6 May 22 2025 21:00:45
%S A218824 1,1,2,9,57,421,3593,34557,366832,4251094,53238166,714702779,
%T A218824 10221402872,154913725486,2477047085038,41629752595369,
%U A218824 732956458329580,13480858878123068,258362762534442843,5148079352377053578,106437899659055825010,2279307634231962670724
%N 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).
%C A218824 Compare the o.g.f. to the LambertW identity:
%C A218824 Sum_{n>=0} n^n * x^n/n! * G(x)^n * exp(-n*x*G(x)) = 1/(1-x*G(x)).
%e A218824 O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 57*x^4 + 421*x^5 + 3593*x^6 +...
%e A218824 such that
%e A218824 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 A218824 where the partition function P(x) = Product_{n>=1} 1/(1-x^n) begins:
%e A218824 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 A218824 (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 A218824 for(n=0,25,print1(a(n),", "))
%Y A218824 Cf. A218670.
%K A218824 nonn
%O A218824 0,3
%A A218824 _Paul D. Hanna_, Nov 06 2012