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A295812 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296170.

%I A295812 #12 Aug 06 2018 08:59:59
%S A295812 1,1,3,19,226,4259,110514,3626207,143043592,6567931068,343278693103,
%T A295812 20092744961109,1300754163383700,92223505422990050,
%U A295812 7104166647498916816,590661172651143976231,52710327177111760030280,5024720072707894279118236,509553454073135435969780828,54771493019290133717304608756,6220332385328132888848047735930,744260531662484056612631555859467
%N A295812 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296170.
%C A295812 E.g.f. G(x) of A296170 satisfies: [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.
%H A295812 Paul D. Hanna, <a href="/A295812/b295812.txt">Table of n, a(n) for n = 1..300</a>
%F A295812 G.f. is the series reversion of the logarithm of the e.g.f. of A296170.
%F A295812 a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = (2 + LambertW(-2*exp(-2)))^2 * sqrt(-LambertW(-2*exp(-2))*(1 + LambertW(-2*exp(-2)))) / (8*sqrt(2)*Pi) = 0.0350943105... - _Vaclav Kotesovec_, Dec 22 2017, updated Aug 06 2018
%e A295812 G.f. A(x) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + 92223505422990050*x^14 + 7104166647498916816*x^15 +...
%e A295812 The series reversion equals the logarithm of the e.g.f. of A296170, which begins:
%e A295812 Series_Reversion(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 +...+ A296171(n)*x^n +...
%o A295812 (PARI) {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff(serreverse(log(Ser(A))),n)}
%o A295812 for(n=1,30,print1(a(n),", "))
%Y A295812 Cf. A296170, A296171, A295813, A295814.
%K A295812 nonn
%O A295812 1,3
%A A295812 _Paul D. Hanna_, Dec 09 2017