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Decagorials: n-th polygorial for k=10.
19

%I #34 May 02 2022 08:13:42

%S 1,1,10,270,14040,1193400,150368400,26314470000,6104957040000,

%T 1813172240880000,670873729125600000,302564051835645600000,

%U 163384587991248624000000,104075982550425373488000000

%N Decagorials: n-th polygorial for k=10.

%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, 10) = (A000142(n)/A000079(n))*A084948(n) = (n!/2^n)*Product_{i=0..n-1} (8*i+2) = (n!/2^n)*8^n*Pochhammer(1/4, n) = (n!/2)*4^n*GAMMA(n+1/4)*sqrt(2)*GAMMA(3/4)/Pi.

%F a(n) = Product_{k=1..n} k*(4k-3). - _Daniel Suteu_, Nov 01 2017

%F D-finite with recurrence a(n) -n*(4*n-3)*a(n-1)=0. - _R. J. Mathar_, May 02 2022

%p a := n->n!/2^n*product(8*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[10, #] &, 14, 0] (* _Robert G. Wilson v_, Dec 26 2016 *)

%o (PARI) a(n)=n!/2^n*prod(i=1,n,8*i-6) \\ _Charles R Greathouse IV_, Dec 13 2016

%Y Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084944, A085356.

%K easy,nonn

%O 0,3

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