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

%I A209903 #20 Oct 26 2024 04:21:42
%S A209903 1,1,4,17,111,752,6893,64171,733540,8751579,119847295,1716294780,
%T A209903 27583937857,460405876777,8428298492136,160944930254405,
%U A209903 3309210789416387,70814345769448444,1617322515279759301,38322855872232745163,960820910852189283072
%N A209903 E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
%H A209903 Seiichi Manyama, <a href="/A209903/b209903.txt">Table of n, a(n) for n = 0..443</a>
%F A209903 E.g.f.: exp( Sum_{n>=1} x^n/n! / (1-x^n) ).
%F A209903 E.g.f.: exp( Sum_{n>=1} A057625(n)*x^n/n! ).
%F A209903 E.g.f.: exp( Sum_{n>=1} exp(x^n)-1 ).
%F A209903 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
%e A209903 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
%e A209903 Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
%e A209903 B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
%e A209903 then the e.g.f. of this sequence equals the infinite product:
%e A209903 A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
%e A209903 The logarithm of the e.g.f. A(x) begins:
%e A209903 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! +...
%o A209903 (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)}
%o A209903 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,x^m/m!/(1-x^m+x*O(x^n)))),n)}
%o A209903 for(n=0,25,print1(a(n),", "))
%o A209903 (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
%Y A209903 Cf. A057625 (log), A209902, A330199.
%K A209903 nonn
%O A209903 0,3
%A A209903 _Paul D. Hanna_, Mar 15 2012