A276482 - OEIS

%I #32 Sep 04 2019 10:05:47

%S 1,1,12,396,25344,2661120,415134720,90084234240,25944259461120,

%T 9573431741153280,4403778600930508800,2470519795122015436800,

%U 1660189302321994373529600,1316530116741341538208972800,1216473827868999581305090867200,1295544626680484554089921773568000

%N a(n) = 5^n*Gamma(n+1/5)*Gamma(n+1)/Gamma(1/5).

%C 12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12).

%C More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2.

%H Robert Israel, <a href="/A276482/b276482.txt">Table of n, a(n) for n = 0..220</a>

%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.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1.

%F a(n) = Product_{k=1..n} A051624(k), a(0)=1.

%F a(n) = A000142(n)*A008548(n).

%F a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)).

%F Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914...

%p seq(mul(k*(5*k-4),k=1..n), n=0..20); # _Robert Israel_, Sep 18 2016

%t FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]]

%t polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2),n]]; Array[polygorial[12, #] &, 16, 0] (* _Robert G. Wilson v_, Dec 13 2016 *)

%o (PARI) a(n) = prod(k=1, n, k*(5*k - 4)); \\ _Michel Marcus_, Sep 06 2016

%Y Cf. A000142, A008548, A051624.

%Y Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11).

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Sep 05 2016