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A034176 One third of quartic factorial numbers. 18

%I

%S 1,7,77,1155,21945,504735,13627845,422463195,14786211825,576662261175,

%T 24796477230525,1165434429834675,59437155921568425,

%U 3269043575686263375,192873570965489539125,12151034970825840964875

%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 M. 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), 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

%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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Last modified January 21 13:54 EST 2020. Contains 331113 sequences. (Running on oeis4.)