A051618 a(n) = (4*n+6)(!^4)/6(!^4).

1, 10, 140, 2520, 55440, 1441440, 43243200, 1470268800, 55870214400, 2346549004800, 107941254220800, 5397062711040000, 291441386396160000, 16903600410977280000, 1048023225480591360000, 69169532881719029760000, 4841867301720332083200000, 358298180327304574156800000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

This sequence is related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).

Row m=6 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

a(n) = A001813 a(n+2)/12. - Zerinvary Lajos, Feb 15 2008

For n>4, a(n) mod n^2 = n*(n-2) if n is prime, otherwise 0. - Gary Detlefs, Apr 16 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617 through A051622 (rows m=0..10).

Programs

Magma

[Factorial(2*n+4)/(12*Factorial(n+2)): n in [0..100]]; // Vincenzo Librandi, Jul 04 2015

Maple

seq(mul((n+2+k), k=1..n+2)/12, n=0..17); # Zerinvary Lajos, Feb 15 2008
A051618 := n -> 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3:
seq(simplify(A051618(n)), n = 0..19);  # Peter Luschny, Jan 22 2025

Mathematica

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 9, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
f[n_] := (2n + 4)!/(12(n + 2)!); Array[f, 16, 0] (* Or *)
FoldList[ #2*#1 &, 1, Range[10, 66, 4]] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[1/(1-4x)^(5/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 24 2015 *)
Table[(Product[(4*k + 6), {k, 0, n}])/6, {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)

Maxima

A051618(n):=(2*n+4)!/(12*(n+2)!)$
makelist(A051618(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

for(n=0,25, print1((2*n+3)!/(6*(n+1)!), ", ")) \\ G. C. Greubel, Jan 27 2017

Formula

a(n) = ((4*n+6)(!^4))/6(!^4).

E.g.f.: 1/(1-4*x)^(5/2).

a(n) = (2n+4)!/(12(n+2)!). - Gary Detlefs, Mar 06 2011

a(n) = (2*n+3)!/(6*(n+1)!). - Gary Detlefs, Apr 16 2012

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

a(n) = (4^(1+n)*Gamma(5/2+n))/(3*sqrt(Pi)). - Gerry Martens, Jul 02 2015

a(n) ~ 2^(2*n+5/2) * n^(n+2) / (3*exp(n)). - Vaclav Kotesovec, Jul 04 2015

a(n) = 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3. - Peter Luschny, Jan 22 2025