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

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

Offset

0,3

Links

Table of n, a(n) for n=0..14.

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A254129 A359926 A334780 * A139564 A264113 A181198

Adjacent sequences: A084938 A084939 A084940 * A084942 A084943 A084944

Keyword

easy,nonn

Author

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

Status

approved