A219184 O.g.f. satisfies: A(x) = Sum_{n>=0} n^(2*n) * x^n * A(x)^n / n! * exp(-n^2*x*A(x)).

1, 1, 8, 112, 2202, 55641, 1724050, 63550446, 2725133134, 133546286188, 7370574862110, 452601918694564, 30610161317492690, 2260721225822606054, 181023122013996360316, 15619416644091171417138, 1444615406376578862379054, 142565035949775130517868740

Offset: 0

Author

Paul D. Hanna, Nov 13 2012

Keywords

• nonn

Comments

Compare to the LambertW identity:

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

Examples

			O.g.f.: A(x) = 1 + x + 8*x^2 + 112*x^3 + 2202*x^4 + 55641*x^5 + 1724050*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(x)^2/2!*exp(-4*x*A(x)) + 3^6*x^3*A(x)^3/3!*exp(-9*x*A(x)) + 4^8*x^4*A(x)^4/4!*exp(-16*x*A(x)) + 5^10*x^5*A(x)^5/5!*exp(-25*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
O.g.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 + 1323652*x^6 +...+ Stirling2(2*n,n)*x^n +...
so that A(x) = (1/x)*Series_Reversion(x/G(x)).
		

Crossrefs

Cf. A007820, A219228, A217900, A218681, A218672.

Programs

PARI

{a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^2*x*A)^m/m!*exp(-m^2*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(2*n,n) * x^n * A(x)^n.