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G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 - x*A(x))^(k+1).
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%I #9 Apr 10 2019 06:16:09

%S 1,2,9,54,379,2948,24736,220622,2074775,20491386,212312349,2310232488,

%T 26473612772,320735694048,4126350096188,56601987176510,

%U 830233489763775,13036492313617494,218958840306428947,3924128327446669670,74779561501535316579,1509296316416028136188

%N G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 - x*A(x))^(k+1).

%F G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000522(k)*x^k*A(x)^k.

%F G.f.: A(x) = (1/x)*Series_Reversion(x/Sum_{k>=0} A000522(k)*x^k).

%F a(n) ~ exp(3) * n!. - _Vaclav Kotesovec_, Apr 10 2019

%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 54*x^3 + 379*x^4 + 2948*x^5 + 24736*x^6 + 220622*x^7 + 2074775*x^8 + 20491386*x^9 + 212312349*x^10 + ...

%t terms = 22; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Floor[Exp[1] k!] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]

%t terms = 22; A[_] = 1; Do[A[x_] = Sum[k! x^k A[x]^k/(1 - x A[x])^(k + 1), {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%t terms = 22; A[_] = 1; Do[A[x_] = 1 + Sum[Floor[Exp[1] k!] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%Y Cf. A000522, A088368, A307441, A307443, A307444.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 08 2019