A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047

Offset

0,3

Comments

Compare to the LambertW identities:

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

(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..50

Example

O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Mathematica

a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)

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)^k/k!*exp(-k*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A218673, A218674, A218675, A218676, A218681.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Cf. A193363, A221409, A221410, A221411, A221412, A221413.

Cf. A209276, A209277.

Sequence in context: A003577 A085928 A130169 * A253109 A167913 A076944

Adjacent sequences: A218669 A218670 A218671 * A218673 A218674 A218675

Keyword

nonn

Author

Paul D. Hanna, Nov 04 2012

Status

approved