A221411 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+2)^n * x^n * A(n*x)^n/n! * exp(-(n+2)*x*A(n*x)).

1, 1, 4, 27, 325, 5553, 140103, 4993445, 253780210, 18321882898, 1882677322704, 275715048156637, 57570654555092091, 17152947168669102772, 7295365645092117304955, 4430848642167010373612127, 3844378527942068170940925685, 4766454891141869269974497298382

Offset

0,3

Comments

Compare to the LambertW identity:

Sum_{n>=0} (n+2)^n * x^n * G(x)^n/n! * exp(-(n+2)*x*G(x)) = 1/(1 - x*G(x)).

Links

Table of n, a(n) for n=0..17.

Example

O.g.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 325*x^4 + 5553*x^5 + 140103*x^6 +...
where
A(x) = exp(-2*x) + 3*x*A(x)*exp(-3*x*A(x)) + 4^2*x^2*A(2*x)^2/2!*exp(-4*x*A(2*x)) + 5^3*x^3*A(3*x)^3/3!*exp(-5*x*A(3*x)) + 6^4*x^4*A(4*x)^4/4!*exp(-6*x*A(4*x)) + 7^5*x^5*A(5*x)^5/5!*exp(-7*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+2)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+2)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A218672, A193363, A221409, A221410, A221412, A221413.

Sequence in context: A336227 A119820 A159599 * A304654 A203202 A320961

Adjacent sequences: A221408 A221409 A221410 * A221412 A221413 A221414

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2013

Status

approved