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Enneagorials: n-th polygorial for k=9.
19

%I #29 May 02 2022 08:03:06

%S 1,1,9,216,9936,745200,82717200,12738448800,2598643555200,

%T 678245967907200,220429939569840000,87290256069656640000,

%U 41375581377017247360000,23128949989752641274240000

%N Enneagorials: n-th polygorial for k=9.

%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.

%F 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).

%F D-finite with recurrence 2*a(n) = n*(7*n-5)*a(n-1). - _R. J. Mathar_, Mar 12 2019

%p a := n->n!/2^n*product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

%t 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 *)

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

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

%K easy,nonn

%O 0,3

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