|
|
A034176
|
|
One third of quartic factorial numbers.
|
|
19
|
|
|
1, 7, 77, 1155, 21945, 504735, 13627845, 422463195, 14786211825, 576662261175, 24796477230525, 1165434429834675, 59437155921568425, 3269043575686263375, 192873570965489539125, 12151034970825840964875, 814119343045331344646625, 57802473356218525469910375
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
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;
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
a(n) = 4^(n-1) * Gamma(n + 3/4) / Gamma(7/4). - Peter McNair, May 06 2024
|
|
MAPLE
|
|
|
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
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|