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%I #31 Dec 18 2022 02:34:28
%S 1,7,77,1155,21945,504735,13627845,422463195,14786211825,576662261175,
%T 24796477230525,1165434429834675,59437155921568425,
%U 3269043575686263375,192873570965489539125,12151034970825840964875,814119343045331344646625,57802473356218525469910375
%N One third of quartic factorial numbers.
%H G. C. Greubel, <a href="/A034176/b034176.txt">Table of n, a(n) for n = 1..350</a>
%H Maxie D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), Article 10.6.7, p 39.
%F 3*a(n) = (4*n-1)(!^4) := product(4*j-1, j=1..n) = (4*n-1)!!/A007696(n) = (4*n)!/(4^n*(2*n)!*A007696(n)), A007696(n)=(4*n-3)(!^4), n >= 1;
%F E.g.f.: (-1 + (1-4*x)^(-3/4))/3.
%F a(n) ~ 4/3*2^(1/2)*Pi^(1/2)*Gamma(3/4)^-1*n^(5/4)*2^(2*n)*e^-n*n^n*{1 + 71/96*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F G.f.: 1/Q(0) where Q(k) = 1 - x + 2*(2*k-1)*x - 4*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F D-finite with recurrence: a(n) +(-4*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 24 2020
%F Sum_{n>=1} 1/a(n) = 3*exp(1/4)*(Gamma(3/4) - Gamma(3/4, 1/4))/sqrt(2). - _Amiram Eldar_, Dec 18 2022
%p A034176:=n->`if`(n=1, 1, (4*n-1)*A034176(n-1)); seq(A034176(n), n=1..20); # _G. C. Greubel_, Aug 15 2019
%t Table[4^n*Pochhammer[3/4, n]/3, {n, 20}] (* _G. C. Greubel_, Aug 15 2019 *)
%o (PARI) m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(4*n-1)*v[n-1]); v \\ _G. C. Greubel_, Aug 15 2019
%o (Magma) [n le 1 select 1 else (4*n-1)*Self(n-1): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019
%o (Sage) [4^n*rising_factorial(3/4, n)/3 for n in (1..20)] # _G. C. Greubel_, Aug 15 2019
%o (GAP) a:=[1];; for n in [2..20] do a[n]:=(4*n-1)*a[n-1]; od; a; # _G. C. Greubel_, Aug 15 2019
%Y Cf. A007696, A034177, A034256, A025749.
%K easy,nonn
%O 1,2
%A _Wolfdieter Lang_
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