A034176 One third of quartic factorial numbers.

1, 7, 77, 1155, 21945, 504735, 13627845, 422463195, 14786211825, 576662261175, 24796477230525, 1165434429834675, 59437155921568425, 3269043575686263375, 192873570965489539125, 12151034970825840964875, 814119343045331344646625, 57802473356218525469910375

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..350

Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, p 39.

Formula

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;

E.g.f.: (-1 + (1-4*x)^(-3/4))/3.

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

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

D-finite with recurrence: a(n) +(-4*n+1)*a(n-1)=0. - R. J. Mathar, Feb 24 2020

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

Maple

A034176:=n->`if`(n=1, 1, (4*n-1)*A034176(n-1)); seq(A034176(n), n=1..20); # G. C. Greubel, Aug 15 2019

Mathematica

Table[4^n*Pochhammer[3/4, n]/3, {n, 20}] (* G. C. Greubel, Aug 15 2019 *)

Prog

(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
(Magma) [n le 1 select 1 else (4*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019
(Sage) [4^n*rising_factorial(3/4, n)/3 for n in (1..20)] # G. C. Greubel, Aug 15 2019
(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

Crossrefs

Cf. A007696, A034177, A034256, A025749.

Sequence in context: A306031 A249933 A107866 * A001765 A246460 A222145

Adjacent sequences: A034173 A034174 A034175 * A034177 A034178 A034179

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved