%I #45 Sep 08 2022 08:44:59

%S 1,10,140,2520,55440,1441440,43243200,1470268800,55870214400,

%T 2346549004800,107941254220800,5397062711040000,291441386396160000,

%U 16903600410977280000,1048023225480591360000,69169532881719029760000,4841867301720332083200000

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

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

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

%C a(n) = A001813 a(n+2)/12. - _Zerinvary Lajos_, Feb 15 2008

%C For n>4, a(n) mod n^2 = n*(n-2) if n is prime, otherwise 0. - _Gary Detlefs_, Apr 16 2012

%H G. C. Greubel, <a href="/A051618/b051618.txt">Table of n, a(n) for n = 0..360</a>

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

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

%F a(n) = (2n+4)!/(12(n+2)!). - _Gary Detlefs_, Mar 06 2011

%F a(n) = (2*n+3)!/(6*(n+1)!). - _Gary Detlefs_, Apr 16 2012

%F 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

%F a(n) = (4^(1+n)*Gamma(5/2+n))/(3*sqrt(Pi)). - _Gerry Martens_, Jul 02 2015

%F a(n) ~ 2^(2*n+5/2) * n^(n+2) / (3*exp(n)). - _Vaclav Kotesovec_, Jul 04 2015

%p seq(mul((n+2+k), k=1..n+2)/12, n=0..17); # _Zerinvary Lajos_, Feb 15 2008

%t s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 9, 5!, 4}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008 *)

%t f[n_] := (2n + 4)!/(12(n + 2)!); Array[f, 16, 0] (* Or *)

%t FoldList[ #2*#1 &, 1, Range[10, 66, 4]] (* _Robert G. Wilson v_ *)

%t With[{nn=20},CoefficientList[Series[1/(1-4x)^(5/2),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 24 2015 *)

%t Table[(Product[(4*k + 6), {k, 0, n}])/6, {n, 0, 50}] (* _G. C. Greubel_, Jan 27 2017 *)

%o (Maxima) A051618(n):=(2*n+4)!/(12*(n+2)!)$

%o makelist(A051618(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (Magma) [Factorial(2*n+4)/(12*Factorial(n+2)): n in [0..100]]; // _Vincenzo Librandi_, Jul 04 2015

%o (PARI) for(n=0,25, print1((2*n+3)!/(6*(n+1)!), ", ")) \\ _G. C. Greubel_, Jan 27 2017

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

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_