A084941 Octagorials: n-th polygorial for k=8.

1, 1, 8, 168, 6720, 436800, 41932800, 5577062400, 981562982400, 220851671040000, 61838467891200000, 21086917550899200000, 8603462360766873600000, 4138265395528866201600000, 2317428621496165072896000000, 1494741460865026472017920000000, 1100129715196659483405189120000000

Offset: 0

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

• easy
• nonn
• changed

Links

• Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Crossrefs

Cf. A006472, A001044, A000680, A073005, A084939, A084940, A084942, A084943, A084944, A085356.

Programs

Maple

a := n->n!/2^n*product(6*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Mathematica

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[8, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

PARI

a(n) = n! / 2^n * prod(i=0, n-1, 6*i+2) \\ Felix Fröhlich, Dec 13 2016

Formula

a(n) = polygorial(n, 8) = (A000142(n)/A000079(n))*A047657(n) = (n!/2^n)*Product_{i=0..n-1} (6*i+2) = (n!/2^n)*6^n*Pochhammer(1/3, n) = (n!/2)*3^n*sqrt(3)*Gamma(n+1/3)*Gamma(2/3)/Pi.

D-finite with recurrence a(n) = n*(3*n-2)*a(n-1). - R. J. Mathar, Mar 12 2019

a(n) ~ 2 * 3^n * n^(2*n + 1/3) * Pi /(Gamma(1/3) * exp(2*n)). - Amiram Eldar, Aug 28 2025