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A295814
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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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3
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1, 7, 591, 360071, 696409901, 2958728428011, 23164541753169117, 300801581861406441263, 6028093825088113213286946, 176753891171734450100762135773, 7275100380834838623971362431809230, 406542590169784279153263825042856310627, 30008177367626616771665421796780382440931316, 2859139755874441545650368872575815286528870509597
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OFFSET
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1,2
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COMMENTS
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E.g.f. G(x) of A296174 satisfies: [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
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LINKS
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FORMULA
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G.f. is the series reversion of the logarithm of the e.g.f. of A296174.
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
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EXAMPLE
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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 +...
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 +...
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! +...
which satisfies [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
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PROG
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(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)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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