OFFSET
0,2
COMMENTS
In general, if k > 1 and g.f. = (1 - (1 - k^k * x)^(1/k)) / (k^(k-1) * x), then a(n) ~ k^(k*n) / (Gamma((k-1)/k) * n^((k+1)/k)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
FORMULA
G.f.: (1 - (1 - 3125*x)^(1/5)) / (625*x).
a(n) ~ 3125^n / (Gamma(4/5) * n^(6/5)).
Recurrence: (n+1)*a(n) = 625*(5*n-1)*a(n-1).
a(n) = 5^(5*n) * Gamma(n+4/5) / (Gamma(4/5) * Gamma(n+2)).
E.g.f.: hypergeom([4/5], [2], 3125*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/5, n+1)*5^(5*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 125^n*A025748(n+1). (End)
MATHEMATICA
CoefficientList[Series[(1-(1-3125*x)^(1/5)) / (625*x), {x, 0, 20}], x]
CoefficientList[Series[Hypergeometric1F1[4/5, 2, 3125*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)
PROG
(Magma) [Round(5^(5*n)*Gamma(n+4/5)/(Gamma(4/5)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022
(SageMath) [5^(5*n)*rising_factorial(4/5, n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jan 27 2015
STATUS
approved