A034301 a(n) = n-th quintic factorial number divided by 4.

1, 9, 126, 2394, 57456, 1666224, 56651616, 2209413024, 97214173056, 4763494479744, 257228701906176, 15176493412464384, 971295578397720576, 67019394909442719744, 4959435223298761261056, 391795382640602139623424, 32910812141810579728367616, 2929062280621141595824717824

Offset

1,2

Links

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

Formula

a(n) = A008546(n)/4.

4*a(n) = (5*n-1)(!^5) = Product_{j=1..n} (5*j-1).

a(n) = (5*n)!/(5^n*n!*A008548(n)*2*A034323(n)*3*A034300(n)).

E.g.f.: (-1 + (1-5*x)^(-4/5))/4, a(0) = 0.

a(n) ~ sqrt(2*Pi) * 5/(4*Gamma(4/5)) * n^(13/10) * (5*n/e)^n * (1 + (241/300)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

D-finite with recurrence: a(n) +(-5*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020

Sum_{n>=1} 1/a(n) = 4*(e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

Maple

a:= n-> mul(5*k-1, k=1..n)/4: seq(a(n), n=1..20); # G. C. Greubel, Aug 23 2019

Mathematica

Table[-5^(n+1)*Pochhammer[-1/5, n+1]/4, {n, 20}] (* G. C. Greubel, Aug 23 2019 *)

Prog

(PARI) a(n) = prod(k=1, n, 5*k-1)/4;
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2019
(Magma) [&*[5*k-1: k in [1..n]]/4: n in [1..20]]; // G. C. Greubel, Aug 23 2019
(Sage) [-5^(n+1)*rising_factorial(-1/5, n+1)/4 for n in (1..20)] # G. C. Greubel, Aug 23 2019
(GAP) List([1..20], n-> Product([1..n], k-> 5*k-1)/4 ); # G. C. Greubel, Aug 23 2019

Crossrefs

Cf. A008546, A008548, A034300, A025750.

Sequence in context: A306033 A065707 A338077 * A092651 A366036 A258294

Adjacent sequences: A034298 A034299 A034300 * A034302 A034303 A034304

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

Status

approved