OFFSET
1,2
COMMENTS
E.g.f. G(x) of A296174 satisfies: [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..160
FORMULA
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
EXAMPLE
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.
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 09 2017
STATUS
approved