A230321 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} LambertW(-k*x)/(-k*x).

1, 1, 3, 19, 203, 3296, 75862, 2340710, 92647529, 4554162028, 271560907586, 19291412245084, 1608220567355569, 155269058419296810, 17162375811312467478, 2150431928188151079196, 302882085165757326494593, 47608431893360236873620584, 8298673711523249659301551906

Offset: 0

Author

Paul D. Hanna, Oct 15 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1746*x^5/5! +...
Let W(x) = LambertW(-x)/(-x), then
W(k*x) = Sum_{j>=0} k^j*(j+1)^(j-1)*x^j/j!
where
A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)/2! + x^3*W(x)*W(2*x)*W(3*x)/3! + x^4*W(x)*W(2*x)*W(3*x)*W(4*x)/4! + x^5*W(x)*W(2*x)*W(3*x)*W(4*x)*W(5*x)/5! +...
RELATED EXPANSIONS:
W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...
W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...
W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...
W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...
...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Crossrefs

Cf. A230320, A230317.

Programs

PARI

{a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n),A=1);
A=sum(m=0,n,x^m/m!*prod(k=1,m,subst(W,x,k*x)));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(A=1);
A=sum(m=0,n,x^m/m!*prod(k=1,m,sum(j=0,n,k^j*(j+1)^(j-1)*x^j/j!)+x*O(x^n) ));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);
A=sum(m=0,n,1/m!^2*prod(k=1,m,subst(-LambertW,x,-k*x)));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

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

E.g.f.: Sum_{n>=0} 1/n!^2 * Product_{k=1..n} -LambertW(-k*x).