A221412 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+3)^n * x^n * A(n*x)^n/n! * exp(-(n+3)*x*A(n*x)).

1, 1, 5, 39, 576, 11693, 358649, 15411564, 951579001, 83392989241, 10419431480203, 1856210104355977, 471536928543684056, 170959559745467848287, 88469465053214549982042, 65371115770077488407503980, 68993903807593031325051425205, 104033290140443202579946504758992

Offset: 0

Author

Paul D. Hanna, Jan 15 2013

Keywords

• nonn

Comments

Compare to the LambertW identity:

Sum_{n>=0} (n+3)^n * x^n * G(x)^n/n! * exp(-(n+3)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 576*x^4 + 11693*x^5 + 358649*x^6 +...
where
A(x) = exp(-3*x) + 4*x*A(x)*exp(-4*x*A(x)) + 5^2*x^2*A(2*x)^2/2!*exp(-5*x*A(2*x)) + 6^3*x^3*A(3*x)^3/3!*exp(-6*x*A(3*x)) + 7^4*x^4*A(4*x)^4/4!*exp(-7*x*A(4*x)) + 8^5*x^5*A(5*x)^5/5!*exp(-8*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218672, A193363, A221409, A221410, A221411, A221413.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+3)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+3)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))