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A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

%I A034385 #28 Apr 02 2024 11:44:38
%S A034385 1,4,40,480,6240,84864,1188096,16972800,246105600,3609548800,
%T A034385 53421322240,796463349760,11946950246400,180123249868800,
%U A034385 2727580640870400,41459225741230080,632253192553758720
%N A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.
%H A034385 Seiichi Manyama, <a href="/A034385/b034385.txt">Table of n, a(n) for n = 0..500</a>
%H A034385 A. Straub, V. H. Moll, and 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 A034385 a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.
%F A034385 G.f.: (1 - 16*x)^(-1/4).
%F A034385 D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - _R. J. Mathar_, Jan 28 2020
%F A034385 From _Peter Bala_, Mar 31 2024: (Start)
%F A034385 a(n) = (-16)^n*binomial(-1/4, n).
%F A034385 a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).
%F A034385 E.g.f.: hypergeom([1/4], [1], 16*x).
%F A034385 a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).
%F A034385 (16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
%F A034385 Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.
%F A034385 Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)
%t A034385 CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* _Harvey P. Dale_, Aug 06 2018 *)
%Y A034385 Cf. A007696.
%Y A034385 Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).
%K A034385 nonn,easy
%O A034385 0,2
%A A034385 _Wolfdieter Lang_