This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A034176 #37 May 28 2024 16:19:48
%S A034176 1,7,77,1155,21945,504735,13627845,422463195,14786211825,576662261175,
%T A034176 24796477230525,1165434429834675,59437155921568425,
%U A034176 3269043575686263375,192873570965489539125,12151034970825840964875,814119343045331344646625,57802473356218525469910375
%N A034176 One third of quartic factorial numbers.
%H A034176 G. C. Greubel, <a href="/A034176/b034176.txt">Table of n, a(n) for n = 1..350</a>
%H A034176 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 A034176 3*a(n) = (4*n-1)(!^4) := Product_{j=1..n} 4*j-1 = (4*n-1)!!/A007696(n) = (4*n)!/(4^n*(2*n)!*A007696(n)), A007696(n)=(4*n-3)(!^4), n >= 1;
%F A034176 E.g.f.: (-1 + (1-4*x)^(-3/4))/3.
%F A034176 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 A034176 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 A034176 D-finite with recurrence: a(n) + (-4*n+1) * a(n-1) = 0. - _R. J. Mathar_, Feb 24 2020
%F A034176 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
%F A034176 a(n) = 4^(n-1) * Gamma(n + 3/4) / Gamma(7/4). - _Peter McNair_, May 06 2024
%p A034176 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 A034176 Table[4^n*Pochhammer[3/4, n]/3, {n, 20}] (* _G. C. Greubel_, Aug 15 2019 *)
%o A034176 (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 A034176 (Magma) [n le 1 select 1 else (4*n-1)*Self(n-1): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019
%o A034176 (Sage) [4^n*rising_factorial(3/4, n)/3 for n in (1..20)] # _G. C. Greubel_, Aug 15 2019
%o A034176 (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 A034176 Cf. A007696, A034177, A034256, A025749.
%K A034176 easy,nonn
%O A034176 1,2
%A A034176 _Wolfdieter Lang_