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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

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Cf. A296172, A296173, A295812, A295814.

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

allocated for Paul DG.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172. Hanna

1, 3, 48, 3271, 575163, 185377116, 93039467356, 66505075585875, 63970743282062646, 79580632411431634441, 124299284968805234137968, 238188439678208173206500760, 549611050835556942751087049225, 1503700734638162443238902233252144, 4814751647416985610768723994195186728, 17841762828286483988438913318683740082187, 75777421917902616009655480827109144353730842

1,2

E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.

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

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 +...

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

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 +...

(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)}

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

Cf. A296172, A296173, A295812.

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nonn

Paul D. Hanna, Dec 09 2017

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