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G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296174.
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%I #13 Oct 13 2020 08:33:52

%S 1,7,591,360071,696409901,2958728428011,23164541753169117,

%T 300801581861406441263,6028093825088113213286946,

%U 176753891171734450100762135773,7275100380834838623971362431809230,406542590169784279153263825042856310627,30008177367626616771665421796780382440931316,2859139755874441545650368872575815286528870509597

%N G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296174.

%C E.g.f. G(x) of A296174 satisfies: [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.

%H Paul D. Hanna, <a href="/A295814/b295814.txt">Table of n, a(n) for n = 1..160</a>

%F G.f. is the series reversion of the logarithm of the e.g.f. of A296174.

%F a(n) ~ sqrt(1-c) * 2^(8*n - 17/2) * n^(3*n - 9/2) / (sqrt(Pi) * c^n * (4-c)^(3*n - 4) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.07930960512711365643910864... - _Vaclav Kotesovec_, Dec 22 2017, updated Oct 13 2020

%e G.f. A(x) = x + 7*x^2 + 591*x^3 + 360071*x^4 + 696409901*x^5 + 2958728428011*x^6 + 23164541753169117*x^7 + 300801581861406441263*x^8 +...

%e Series_Reversion(A(x)) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 +...+ A296175(n)*x^n +...

%e G(x) = exp(Series_Reversion(A(x))) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! +...+ A296174(n)*x^n/n! +...

%e which satisfies [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.

%o (PARI) {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff(serreverse(log(Ser(A))),n)}

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

%Y Cf. A296174, A296175, A295812, A295813.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 09 2017