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

%I A034688 #33 Sep 24 2023 09:15:55
%S A034688 1,5,75,1375,27500,577500,12512500,277062500,6233906250,141994531250,
%T A034688 3265874218750,75708902343750,1766541054687500,41445770898437500,
%U A034688 976936028320312500,23120819336914062500,549119459251708984375
%N A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548.
%H A034688 Seiichi Manyama, <a href="/A034688/b034688.txt">Table of n, a(n) for n = 0..500</a>
%H A034688 A. Straub, V. H. Moll, T. Amdeberhan, <a href="http://dx.doi.org/10.4064/aa140-1-2">The p-adic valuation of k-central binomial coefficients</a>, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)
%F A034688 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).
%F A034688 G.f.: (1-25*x)^(-1/5).
%F A034688 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
%F A034688 a(n) = (-25)^n*binomial(-1/5, n). - _Peter Luschny_, Oct 23 2018
%F A034688 E.g.f.: L_{-1/5}(25*x), where L_{k}(x) is the Laguerre polynomial. - _Stefano Spezia_, Aug 17 2019
%F A034688 D-finite with recurrence: n*a(n) +5*(-5*n+4)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%p A034688 A034688 := n -> (-25)^n*binomial(-1/5, n):
%p A034688 seq(A034688(n), n=0..16); # _Peter Luschny_, Oct 23 2018
%t A034688 Table[(-25)^n*Binomial[-1/5,n], {n,0,20}] (* _G. C. Greubel_, Aug 17 2019 *)
%t A034688 CoefficientList[Series[1/Surd[1-25x,5],{x,0,20}],x] (* _Harvey P. Dale_, Sep 11 2022 *)
%o A034688 (PARI) vector(20, n, n--; 5^n*prod(k=0, n-1, 5*k+1)/n!) \\ _G. C. Greubel_, Aug 17 2019
%o A034688 (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
%o A034688 (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
%o A034688 (GAP) List([0..20], n-> 5^n*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # _G. C. Greubel_, Aug 17 2019
%Y A034688 Cf. A008548, A034385, A034687, A049380, A049381, A049382.
%K A034688 easy,nonn
%O A034688 0,2
%A A034688 _Wolfdieter Lang_