A084944 Hendecagorials: n-th polygorial for k=11.

1, 1, 11, 330, 19140, 1818300, 256380300, 50250538800, 13065140088000, 4350691649304000, 1805537034461160000, 913601739437346960000, 553642654099032257760000, 395854497680808064298400000, 329746796568113117560567200000, 316556924705388592858144512000000, 346946389477105897772526385152000000

Offset: 0

Author

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

Keywords

• easy
• nonn
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 0..220
• Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A085356.

Programs

Maple

a := n->n!/2^n*product(9*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[11, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)

Formula

a(n) = polygorial(n, 11) = (A000142(n)/A000079(n))*A084949(n) = (n!/2^n)*Product_{i=0..n-1} (9*i+2) = (n!/2^n)*9^n*Pochhammer(2/9, n) = (n!/2^n)*9^n*Gamma(n+2/9)/Gamma(2/9).

D-finite with recurrence 2*a(n) = n*(9*n-7)*a(n-1). - R. J. Mathar, Mar 12 2019

a(n) ~ 9^n * n^(2*n + 2/9) * Pi /(Gamma(2/9) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025