A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548.

1, 5, 75, 1375, 27500, 577500, 12512500, 277062500, 6233906250, 141994531250, 3265874218750, 75708902343750, 1766541054687500, 41445770898437500, 976936028320312500, 23120819336914062500, 549119459251708984375

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)

Formula

a(n) = (5^n/n!)*A008548(n), n >= 1, a(0) := 1, where A008548(n)=(5*n-4)(!^5) := Product_{j=1..n} (5*j-4).

G.f.: (1-25*x)^(-1/5).

a(n) ~ Gamma(1/5)^-1*n^(-4/5)*5^(2*n)*{1 - 2/25*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

a(n) = (-25)^n*binomial(-1/5, n). - Peter Luschny, Oct 23 2018

E.g.f.: L_{-1/5}(25*x), where L_{k}(x) is the Laguerre polynomial. - Stefano Spezia, Aug 17 2019

D-finite with recurrence: n*a(n) +5*(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

Maple

A034688 := n -> (-25)^n*binomial(-1/5, n):
seq(A034688(n), n=0..16); # Peter Luschny, Oct 23 2018

Mathematica

Table[(-25)^n*Binomial[-1/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 17 2019 *)
CoefficientList[Series[1/Surd[1-25x, 5], {x, 0, 20}], x] (* Harvey P. Dale, Sep 11 2022 *)

Prog

(PARI) vector(20, n, n--; 5^n*prod(k=0, n-1, 5*k+1)/n!) \\ G. C. Greubel, Aug 17 2019
(Magma) [1] cat [5^n*(&*[5*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 17 2019
(Sage) [5^n*product(5*k+1 for k in (0..n-1))/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 17 2019
(GAP) List([0..20], n-> 5^n*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # G. C. Greubel, Aug 17 2019

Crossrefs

Cf. A008548, A034385, A034687, A049380, A049381, A049382.

Sequence in context: A224088 A219462 A091882 * A238608 A132855 A238560

Adjacent sequences: A034685 A034686 A034687 * A034689 A034690 A034691

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved