A276482 a(n) = 5^n*Gamma(n+1/5)*Gamma(n+1)/Gamma(1/5).

1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000

Offset: 0

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

• nonn
• easy

Comments

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

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.

Links

• Robert Israel, Table of n, a(n) for n = 0..220
• Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A008548, A051624.

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).

Programs

Maple

seq(mul(k*(5*k-4),k=1..n), n=0..20); # Robert Israel, Sep 18 2016

Mathematica

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

PARI

a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016

Formula

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

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

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

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

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...