A084942 Enneagorials: n-th polygorial for k=9.

1, 1, 9, 216, 9936, 745200, 82717200, 12738448800, 2598643555200, 678245967907200, 220429939569840000, 87290256069656640000, 41375581377017247360000, 23128949989752641274240000, 15056946443328969469530240000, 11292709832496727102147680000000, 9666559616617198399438414080000000

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, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

Programs

Maple

a := n->n!/2^n*product(7*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[9, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

PARI

a(n)=n!/2^n*prod(i=1,n,7*i-5) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = polygorial(n, 9) = (A000142(n)/A000079(n))*A084947(n) = (n!/2^n)*Product_{i=0..n-1} (7*i+2) = (n!/2^n)*7^n*Pochhammer(2/7, n) = (n!/2^n)*7^n*Gamma(n+2/7)/Gamma(2/7).

D-finite with recurrence 2*a(n) = n*(7*n-5)*a(n-1). - R. J. Mathar, Mar 12 2019

a(n) ~ 7^n * n^(2*n + 2/7) * Pi /(Gamma(2/7) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025