A295813 - OEIS

%I #8 Oct 13 2020 11:51:09

%S 1,3,48,3271,575163,185377116,93039467356,66505075585875,

%T 63970743282062646,79580632411431634441,124299284968805234137968,

%U 238188439678208173206500760,549611050835556942751087049225,1503700734638162443238902233252144,4814751647416985610768723994195186728,17841762828286483988438913318683740082187,75777421917902616009655480827109144353730842

%N G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172.

%C 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 Paul D. Hanna, <a href="/A295813/b295813.txt">Table of n, a(n) for n = 1..200</a>

%F G.f. is the series reversion of the logarithm of the e.g.f. of A296172.

%F 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 G.f.: 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 +...

%e The series reversion equals the logarithm of the e.g.f. of A296172, which begins:

%e Series_Reversion(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 +...+ A296173(n)*x^n +...

%o (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(serreverse(log(Ser(A))), n)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A296172, A296173, A295812, A295814.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 09 2017