A386648 E.g.f. A(x) satisfies 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.

1, 1, 10, 63, 806, 9485, 161540, 2752155, 59021506, 1310350929, 34127883032, 934203381287, 28851653188814, 942891341070237, 33962521081521076, 1297690184864525043, 53814377103189792794, 2366017294084046632937, 111607600081334119137488, 5565256312162642805357247

Offset: 1

Author

Paul D. Hanna, Aug 12 2025

Keywords

• nonn

Comments

Conjecture: for n > 6, a(n) (mod 6) equals [2,3,4,3,2,5] repeating.

In general, the following sums are equal:

(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = A(x) with p = LambertW(-x), r = 1.

Examples

			E.g.f.: A(x) = x + x^2/2! + 10*x^3/3! + 63*x^4/4! + 806*x^5/5! + 9485*x^6/6! + 161540*x^7/7! + 2752155*x^8/8! + 59021506*x^9/9! + 1310350929*x^10/10! + ...
where 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
RELATED SERIES.
-LambertW(-x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/6! + 7^6*x^7/7! + ... + n^(n-1)*x^n/n! + ...
where exp(LambertW(-x)) = -x/LambertW(-x);
also, (-x/LambertW(-x))^y = Sum_{k>=0} y*(y - k)^(k-1) * (-x)^k/k!.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..102

Crossrefs

Cf. A386645.

Programs

PARI

{a(n) = my(A=[0,1]); for(i=0, n, A=concat(A, 0);
A[#A] = polcoeff(1 - sum(m=0, #A, (Ser(A)^m + lambertw(-x +x^3*O(x^n)))^m /m! ), #A-1) ); n!*A[n+1]}
for(n=1, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.

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

(3) 1 = Sum_{n>=0} A(x)^(n^2) * (-x/LambertW(-x))^(A(x)^n) / n!.

(4) 1 = Sum_{n>=0} A(x)^(n*(n+1))/n! * Sum_{k>=0} (A(x)^n - k)^(k-1) * (-x)^k/k!.