A344218 Expansion of e.g.f. log(1 + (1/(1-x)^5 - 1)/5).

0, 1, 5, 26, 126, 408, -1704, -51696, -555408, -1217664, 93550464, 2424183552, 30038190336, -114098181120, -16707096806400, -459530721441792, -5231858686838784, 130925278326915072, 9038174050387722240, 246578101419998380032, 1534994756662100557824

Offset: 0

Author

Vaclav Kotesovec, May 12 2021

Keywords

• sign

Comments

In general, column k > 2 of A308497 is asymptotic to -2*(n-1)! * cos(n*arctan(sin(Pi/k)/(cos(Pi/k) - (k-1)^(1/k)))) / (1 + 1/(k-1)^(2/k) - 2*cos(Pi/k)/(k-1)^(1/k))^(n/2).

Crossrefs

Column k=5 of A308497.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Log[1 + (1/(1 - x)^5 - 1)/5], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) ~ -2*(n-1)! * cos(n*arctan(5^(1/4) / (phi^(1/2)*(phi - 2^(7/5))))) / (1 + 1/2^(4/5) - phi/2^(2/5))^(n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.