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

0,3

Links

Table of n, a(n) for n=0..16.

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

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

From Amiram Eldar, Dec 26 2025: (Start)

Sum_{n>=0} 1/a(n) = (2/7)^(5/14) * BesselI(-5/7, 2*sqrt(2/7)) * Gamma(2/7).

Sum_{n>=0} (-1)^n/a(n) = (2/7)^(5/14) * BesselJ(-5/7, 2*sqrt(2/7)) * Gamma(2/7). (End)

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 *)

Prog

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

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

Cf. A395756 (sum of reciprocals).

Sequence in context: A007107 A217042 A064633 * A061718 A085741 A211044

Adjacent sequences: A084939 A084940 A084941 * A084943 A084944 A084945

Keyword

easy,nonn

Author

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

Status

approved