A326266 - OEIS

A326266

E.g.f.: Sum_{n>=0} (W(x)^n - 1)^n / n!, where W(x) = exp(x*W(x)) = LambertW(-x)/(-x).

1, 1, 7, 91, 1783, 47946, 1672792, 72866697, 3852230053, 241824521557, 17714982044177, 1493077817195504, 143094233569327124, 15440409366381056045, 1860025278971873645275, 248329234183480721887287, 36510264273068226851851499, 5878143072506946449089361730, 1031187834682741732109817310932, 196233233091044380685807479720997, 40346356057197038193312451911514301

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OFFSET

0,3

COMMENTS

More generally, the following sums are equal:

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

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

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

LINKS

Table of n, a(n) for n=0..20.

FORMULA

Let W(x) = LambertW(-x)/(-x), then e.g.f. A(x) equals the following sums.

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

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 91*x^3/3! + 1783*x^4/4! + 47946*x^5/5! + 1672792*x^6/6! + 72866697*x^7/7! + 3852230053*x^8/8! + 241824521557*x^9/9! + 17714982044177*x^10/10! + ...

such that

A(x) = 1 + (W(x) - 1) + (W(x)^2 - 1)^2/2! + (W(x)^3 - 1)^3/3! + (W(x)^4 - 1)^4/4! + (W(x)^5 - 1)^5/5! + (W(x)^6 - 1)^6/6! + (W(x)^7 - 1)^7/7! + ...

also

A(x) = exp(-1) + W(x)*exp(-W(x)) + W(x)^4*exp(-W(x)^2)/2! + W(x)^9*exp(-W(x)^3)/3! + W(x)^16*exp(-W(x)^4)/4! + W(x)^25*exp(-W(x)^5)/5! + W(x)^36*exp(-W(x)^6)/6! + ...

where W(x) = exp(x*W(x)) = LambertW(-x)/(-x) begins

W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! + ... + (n+1)^(n-1)*x^n/n! + ...

RELATED SERIES.

Note that W(x)^n equals

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

and so

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

PROG

(PARI) /* E.g.f.: Sum_{n>=0} (W(x)^n - 1)^n / n! */

{a(n) = my(W = 1/x*serreverse(x*exp(-x +x*O(x^n))));

n! * polcoeff( sum(m=0, n, (W^m - 1)^m / m!), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A326267, A326268.