cp's OEIS Frontend

A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).

%I A355843 #26 Mar 04 2024 04:28:47
%S A355843 1,0,2,3,40,185,2556,22057,349616,4519377,83642860,1439639201,
%T A355843 31015493928,663158322697,16468280168900,418772642545545,
%U A355843 11847925722273376,348085509493265825,11091199095506163420,368912674236287743633,13099432280183074041560
%N A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).
%F A355843 E.g.f.: exp( -LambertW(x * (1 - exp(x))) ).
%F A355843 E.g.f.: LambertW(x * (1 - exp(x))) / (x * (1 - exp(x))).
%F A355843 a(n) ~ sqrt(1 + exp(1+r)*r^2) * n^(n-1) / (exp(n-1) * r^n), where r = 0.528399250336668412340528181936966763473482889289226687323... is the root of the equation exp(1+r) - exp(1) = 1/r. - _Vaclav Kotesovec_, Jul 21 2022
%F A355843 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - _Seiichi Manyama_, Aug 28 2022
%t A355843 nmax = 20; A[_] = 1;
%t A355843 Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A355843 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A355843 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x*(1-exp(x))))))
%o A355843 (PARI) a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ _Seiichi Manyama_, Aug 28 2022
%Y A355843 Cf. A052880, A355781, A356785.
%K A355843 nonn
%O A355843 0,3
%A A355843 _Seiichi Manyama_, Jul 18 2022