A211398 E.g.f.: Sum_{n>=0} log( Sum_{k>=0} (n+k)^k*x^k/k! )^n / n!.

1, 2, 14, 153, 2262, 42120, 945823, 24870937, 749702348, 25490350284, 965219424913, 40287663094503, 1837912330721162, 90988147658574582, 4858595700832370019, 278371180944699911227, 17034913752075240500920, 1108950617553341656112312, 76523438074756638449399565

Offset

0,2

Comments

Compare to the trivial identities:

(1) Sum_{n>=0} log( Sum_{k>=0} n^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!;

(2) Sum_{n>=0} log( Sum_{k>=0} k^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!.

Links

Table of n, a(n) for n=0..18.

Formula

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

E.g.f.: Sum_{n>=0} [ Sum_{k>=1} (k^(k-1)*n + A001865(k))*x^k/k! ]^n / n!.

Example

E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 153*x^3/3! + 2262*x^4/4! + 42120*x^5/5! +...
such that
A(x) = 1 + log(F(x,1)) + log(F(x,2))^2/2! + log(F(x,3))^3/3! + log(F(x,4))^4/4! +...
where
F(x,n) = 1 + (n+1)*x + (n+2)^2*x^2/2! + (n+3)^3*x^3/3! + (n+4)^4*x^4/4! + (n+5)^5*x^5/5! +...+ (n+k)^k*x^k/k! +...
Also,
A(x) = 1 + G(x,1) + G(x,2)^2/2! + G(x,3)^3/3! + G(x,4)^4/4! +...+ G(x,n)^n/n! +...
where G(x,n) = log( (LambertW(-x)/(-x))^n / (1+LambertW(-x)) ):
G(x,n) = (n+1)*x + (2*n+3)*x^2/2! + (9*n+17)*x^3/3! + (64*n+142)*x^4/4! + (625*n+1569)*x^5/5! +...+ (k^(k-1)*n + A001865(k))*x^k/k! +...
Related expansion:
Sum_{n>=0} n^n*log(LambertW(-x)/(-x))^n/n! = 1/(1+LambertW(LambertW(-x)));
1/(1+LambertW(LambertW(-x))) = 1 + x + 6*x^2/2! + 60*x^3/3! + 836*x^4/4! +...

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, log(sum(k=0, n, (m+k)^k*x^k/k! +x*O(x^n)))^m/m!), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000169, A001865.

Sequence in context: A121227 A250916 A354751 * A219430 A301931 A060193

Adjacent sequences: A211395 A211396 A211397 * A211399 A211400 A211401

Keyword

nonn

Author

Paul D. Hanna, Feb 10 2013

Status

approved