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A034176 One third of quartic factorial numbers. 18
1, 7, 77, 1155, 21945, 504735, 13627845, 422463195, 14786211825, 576662261175, 24796477230525, 1165434429834675, 59437155921568425, 3269043575686263375, 192873570965489539125, 12151034970825840964875 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 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

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

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Last modified December 13 17:17 EST 2019. Contains 329970 sequences. (Running on oeis4.)