A034724 a(n) = n-th sextic factorial number divided by 4.

1, 10, 160, 3520, 98560, 3351040, 134041600, 6165913600, 320627507200, 18596395417600, 1190169306726400, 83311851470848000, 6331700711784448000, 519199458366324736000, 45689552336236576768000, 4294817919606238216192000, 429481791960623821619200000

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 1..345

Crossrefs

Cf. A034000, A008542, A034689, A073006.

Programs

GAP

List([1..20], n-> Product([1..n], j-> 6*j-2)/4 ); # G. C. Greubel, Nov 11 2019

Magma

[(&*[6*j-2: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Nov 11 2019

Maple

seq( mul(6*j-2, j=1..n)/4, n=1..20); # G. C. Greubel, Nov 11 2019

Mathematica

With[{nn=20},CoefficientList[Series[((1-6x)^(-2/3)-1)/4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 02 2017 *)
Table[6^n*Pochhammer[2/3, n]/4, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

PARI

vector(20, n, prod(j=1,n, 6*j-2)/4 ) \\ G. C. Greubel, Nov 11 2019

Sage

[product( (6*j-2) for j in (1..n))/4 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

4*a(n) = (6*n-2)(!^6) = Product_{j=1..n} (6*j-2).

a(n) = 2^(n+1)*A034000(n), 2*A034000(n) = (3*n-1)(!^3).

E.g.f.: (-1 + (1-6*x)^(-2/3))/4.

D-finite with recurrence: a(n) +2*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Jan 28 2020

Sum_{n>=1} 1/a(n) = 4*(e/6^2)^(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6)). - Amiram Eldar, Dec 18 2022

a(n) ~ sqrt(Pi) * 2^(n-3/2) * (3/e)^n * n^(n+1/6) / Gamma(2/3). - Amiram Eldar, Sep 01 2025