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

0,3

Links

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

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

Example

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! +...

Prog

(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

Crossrefs

Cf. A057625 (log), A209902, A330199.

Sequence in context: A232211 A122940 A077386 * A330514 A330535 A004140

Adjacent sequences: A209900 A209901 A209902 * A209904 A209905 A209906

Keyword

nonn

Author

Paul D. Hanna, Mar 15 2012

Status

approved