A254287 Expansion of (1 - (1 - 3125*x)^(1/5)) / (625*x).

1, 1250, 2343750, 5126953125, 12176513671875, 30441284179687500, 78821182250976562500, 209368765354156494140625, 567040406167507171630859375, 1559361116960644721984863281250, 4341403109719976782798767089843750, 12210196246087434701621532440185546875

Offset: 0

Author

Vaclav Kotesovec, Jan 27 2015

Keywords

• nonn
• easy

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

Crossrefs

Cf. A000108 (k=2), A254282 (k=3), A254286 (k=4).

Cf. A008546, A025748.

Programs

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

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 *)

SageMath

[5^(5*n)*rising_factorial(4/5, n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

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)