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A199163 E.g.f.: Product_{n>=1} 1/(1 - x^n/n!)^(2^n).

%I A199163 #14 Jul 21 2019 09:17:15
%S A199163 1,2,10,56,404,3272,30688,316304,3609656,44624624,597131600,
%T A199163 8555752064,130941724064,2127005759296,36577172243776,663151898857856,
%U A199163 12646212723268576,252922802230850496,5294480128736395456,115747660924409952512,2638244919080808844544
%N A199163 E.g.f.: Product_{n>=1} 1/(1 - x^n/n!)^(2^n).
%C A199163 Compare to the e.g.f. of A005651 (sum of multinomial coefficients): Product_{n>=1} 1/(1 - x^n/n!).
%F A199163 a(n) ~ c * n! * n, where c = Product_{k>=2} (1 - 1/k!)^(2^k) = 200.80872040834836274... - _Vaclav Kotesovec_, Jul 21 2019
%e A199163 E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 56*x^3/3! + 404*x^4/4! + 3272*x^5/5! +...
%e A199163 where
%e A199163 A(x) = 1/((1-x)^2*(1-x^2/2!)^4*(1-x^3/3!)^8*(1-x^4/4!)^16*(1-x^5/5!)^32*...).
%t A199163 nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/k!)^(2^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 21 2019 *)
%o A199163 (PARI) {a(n)=n!*polcoeff(prod(k=1, n, 1/(1-x^k/k!+x*O(x^n))^(2^k)), n)}
%Y A199163 Cf. A005651.
%K A199163 nonn
%O A199163 0,2
%A A199163 _Paul D. Hanna_, Nov 09 2011