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A296175 G.f. equals the logarithm of the e.g.f. of A296174. 6

%I #10 Oct 13 2020 08:37:22

%S 1,-7,-493,-341101,-680813601,-2923660883625,-22996362478599551,

%T -299331006952284448127,-6006951481145880962408552,

%U -176288642409787912257773903552,-7260231964238768891891716773249396,-405879958110794676900559524931590299892,-29968312587171485511980894312242331299164248,-2855987647850204274493781603297327940940773633392

%N G.f. equals the logarithm of 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="/A296175/b296175.txt">Table of n, a(n) for n = 1..160</a>

%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_, Oct 13 2020

%e G.f. 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 - 176288642409787912257773903552*x^10 - 7260231964238768891891716773249396*x^11 - 405879958110794676900559524931590299892*x^12 +...

%e such that

%e G(x) = exp(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! - 639738016495616440994202167765715629*x^10/10! +...

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

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

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

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

%Y Cf. A296174, A296171, A296173, A296177.

%K sign

%O 1,2

%A _Paul D. Hanna_, Dec 07 2017

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)