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%I #65 May 02 2022 08:02:05
%S 1,1,5,60,1320,46200,2356200,164934000,15173928000,1775349576000,
%T 257425688520000,45306921179520000,9514453447699200000,
%U 2350070001581702400000,674470090453948588800000,222575129849803034304000000
%N Pentagorials: n-th polygorial for k=5.
%H Robert Israel, <a href="/A084939/b084939.txt">Table of n, a(n) for n = 0..243</a>
%H M. A. Asiru, <a href="http://dx.doi.org/10.1080/0020739X.2012.729684">Sequence factorial of g-gonal numbers</a>, Int. J. Math. Educ. Sci. Technol., 44(4) (2012), 579-586.
%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.
%F a(n) = polygorial(n, 5) = (A000142(n)/A000079(n))*A008544(n) = (n!/2^n)*Product_{i=0..n-1} (3*i+2) = (n!/2^n)*3^n*Pochhammer(2/3, n) = (n!/2^n)*3^n*GAMMA(n+2/3)/GAMMA(2/3).
%F a(n) ~ Gamma(1/3) * 3^(n+1/2) * n^(2*n+2/3) / (2^n * exp(2*n)). - _Vaclav Kotesovec_, Jul 17 2015
%F D-finite with recurrence a(n+1) = ((n+1)*(3*n+2)/2)*a(n) = A000326(n+1)*a(n). - _Muniru A Asiru_, Apr 05 2016
%F E.g.f.: hypergeom([2/3, 1], [], (3/2)*x). - _Robert Israel_, Apr 05 2016
%p a := n->(n!/2^n)*mul(3*i+2,i=0..n-1); [seq(a(j),j=0..30)];
%t Table[k! Pochhammer[2/3, k] (3/2)^k, {k, 0, 20}] (* _Jan Mangaldan_, Mar 20 2013 *)
%t polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[5, #] &, 17, 0] (* _Robert G. Wilson v_, Dec 17 2016 *)
%o (PARI) a(n)=n!/2^n*prod(i=1,n,3*i-1) \\ _Charles R Greathouse IV_, Dec 13 2016
%Y Cf. A000326, A006472, A001044, A000680, A084940, A084941, A084942, A084943, A084944, A085356.
%K easy,nonn
%O 0,3
%A Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
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