A209903 E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

1, 1, 4, 17, 111, 752, 6893, 64171, 733540, 8751579, 119847295, 1716294780, 27583937857, 460405876777, 8428298492136, 160944930254405, 3309210789416387, 70814345769448444, 1617322515279759301, 38322855872232745163, 960820910852189283072

Offset: 0

Author

Paul D. Hanna, Mar 15 2012

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
then the e.g.f. of this sequence equals the infinite product:
A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
The logarithm of the e.g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2! + 7*x^3/3! + 37*x^4/4! + 121*x^5/5! + 1201*x^6/6! +...+ A057625(n)*x^n/n! +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..443

Crossrefs

Cf. A057625 (log), A209902, A330199.

Programs

PARI

{a(n)=local(Bell=exp(exp(x+x*O(x^n))-1));n!*polcoeff(prod(m=1,n,subst(Bell,x,x^m+x*O(x^n))),n)}

PARI

{a(n)=n!*polcoeff(exp(sum(m=1,n,x^m/m!/(1-x^m+x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))

PARI

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/d!)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Jul 02 2021

Formula

E.g.f.: exp( Sum_{n>=1} x^n/n! / (1-x^n) ).

E.g.f.: exp( Sum_{n>=1} A057625(n)*x^n/n! ).

E.g.f.: exp( Sum_{n>=1} exp(x^n)-1 ).

a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/d!) * a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jul 02 2021