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A296173 G.f. equals the logarithm of the e.g.f. of A296172.

%I A296173 #14 Oct 13 2020 11:59:29
%S A296173 1,-3,-30,-2686,-517311,-173118807,-88535206152,-63977172334344,
%T A296173 -61971659588102940,-77470793599569049440,-121439997599825393413344,
%U A296173 -233353875172602479932391040,-539638027429765922735002220880,-1479049138515818646669055218090480,-4742815067612592169849894663392228480,-17597031102801426396121130730318359114880,-74817150772352720408567833273371047298417408
%N A296173 G.f. equals the logarithm of the e.g.f. of A296172.
%C A296173 E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.
%H A296173 Paul D. Hanna, <a href="/A296173/b296173.txt">Table of n, a(n) for n = 1..180</a>
%F A296173 a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(2*n - 7/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 3) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - _Vaclav Kotesovec_, Oct 13 2020
%e A296173 G.f. A(x) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...
%e A296173 such that
%e A296173 G(x) = exp(A(x)) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
%e A296173 satisfies [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.
%e A296173 Series_Reversion(A(x)) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...+ A295813(n)*x^n +...
%o A296173 (PARI) {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(log(Ser(A)),n)}
%o A296173 for(n=1,30,print1(a(n),", "))
%Y A296173 Cf. A296172, A295813, A296171, A296175, A296177.
%K A296173 sign
%O A296173 1,2
%A A296173 _Paul D. Hanna_, Dec 07 2017