OFFSET
0,3
LINKS
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
FORMULA
a(n) = polygorial(n, 8) = (A000142(n)/A000079(n))*A047657(n) = (n!/2^n)*Product_{i=0..n-1} (6*i+2) = (n!/2^n)*6^n*Pochhammer(1/3, n) = (n!/2)*3^n*sqrt(3)*Gamma(n+1/3)*Gamma(2/3)/Pi.
D-finite with recurrence a(n) = n*(3*n-2)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 2 * 3^n * n^(2*n + 1/3) * Pi /(Gamma(1/3) * exp(2*n)). - Amiram Eldar, Aug 28 2025
Sum_{n>=0} 1/a(n) = (3^(1/3)/2) * Gamma(1/3) * (AiryBi'(3^(1/3))/sqrt(3) - AiryAi'(3^(1/3))), where AiryAi' and AiryBi' are the derivatives of the Airy functions of the first and second kind, respectively. - Amiram Eldar, Dec 26 2025
MAPLE
a := n->n!/2^n*product(6*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[8, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
PROG
(PARI) a(n) = n! / 2^n * prod(i=0, n-1, 6*i+2) \\ Felix Fröhlich, Dec 13 2016
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
STATUS
approved
