A223075 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A((2*n-1)*x)^n/n! * exp(-n*x*A((2*n-1)*x)).

1, 1, 2, 13, 132, 2492, 76726, 4048401, 360486616, 54950141846, 14338767268684, 6424397920197266, 4947731418324541980, 6554636080888858780850, 14947781374271898418583534, 58699996835841575449007944393, 397110307362512858324163841229032

Offset

0,3

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)).

Links

Table of n, a(n) for n=0..16.

Formula

a(n) == 1 (mod 2) when n = 2^m-1 for m>=0, and a(n) == 0 (mod 2) otherwise.

Example

O.g.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 132*x^4 + 2492*x^5 + 76726*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(3*x)^2/2!*exp(-2*x*A(3*x)) + 3^3*x^3*A(5*x)^3/3!*exp(-3*x*A(5*x)) + 4^4*x^4*A(7*x)^4/4!*exp(-4*x*A(7*x)) + 5^5*x^5*A(9*x)^5/5!*exp(-5*x*A(9*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^k*x^k*subst(A, x, (2*k-1)*x)^k/k!*exp(-k*x*subst(A, x, (2*k-1)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A223076, A218672, A217900.

Sequence in context: A085882 A041509 A079165 * A239748 A355113 A065132

Adjacent sequences: A223072 A223073 A223074 * A223076 A223077 A223078

Keyword

nonn

Author

Paul D. Hanna, Mar 14 2013

Status

approved