A216246 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x^2)^n/n! * exp(-n*x*A(n*x^2)).

1, 1, 1, 2, 7, 20, 62, 228, 867, 3474, 14864, 67101, 316028, 1540782, 7792810, 40669011, 218087867, 1201778736, 6792071574, 39309770969, 232718797430, 1407837227275, 8694994837673, 54771751869237, 351643945379956, 2299330292987022, 15302662859459784

Offset

0,4

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..26.

Example

O.g.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 20*x^5 + 62*x^6 + 228*x^7 +...
where
A(x) = 1 + x*A(x^2)*exp(-x*A(x^2)) + 2^2*x^2*A(2*x^2)^2/2!*exp(-2*x*A(2*x^2)) + 3^3*x^3*A(3*x^2)^3/3!*exp(-3*x*A(3*x^2)) + 4^4*x^4*A(4*x^2)^4/4!*exp(-4*x*A(4*x^2)) + 5^5*x^5*A(5*x^2)^5/5!*exp(-5*x*A(5*x^2)) +...
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, k*x^2)^k/k!*exp(-k*x*subst(A, x, k*x^2)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A216247, A218672, A217900.

Sequence in context: A015518 A014983 A083379 * A322202 A000935 A035071

Adjacent sequences: A216243 A216244 A216245 * A216247 A216248 A216249

Keyword

nonn

Author

Paul D. Hanna, Mar 14 2013

Status

approved