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A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548. 12
1, 5, 75, 1375, 27500, 577500, 12512500, 277062500, 6233906250, 141994531250, 3265874218750, 75708902343750, 1766541054687500, 41445770898437500, 976936028320312500, 23120819336914062500, 549119459251708984375 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
Sequence in context: A224088 A219462 A091882 * A238608 A132855 A238560
KEYWORD
easy,nonn
AUTHOR
STATUS
approved

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Last modified April 12 16:59 EDT 2024. Contains 371635 sequences. (Running on oeis4.)