A084943 Decagorials: n-th polygorial for k=10.

1, 1, 10, 270, 14040, 1193400, 150368400, 26314470000, 6104957040000, 1813172240880000, 670873729125600000, 302564051835645600000, 163384587991248624000000, 104075982550425373488000000, 77224379052415627128096000000, 66026844089815361194522080000000, 64442199831659792525853550080000000

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, 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.

a(n) = Product_{k=1..n} k*(4k-3). - Daniel Suteu, Nov 01 2017

D-finite with recurrence a(n) - n*(4*n-3)*a(n-1) = 0. - R. J. Mathar, May 02 2022

a(n) ~ 2^(2*n+1) * n^(2*n + 1/4) * Pi /(Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 28 2025

From Amiram Eldar, Dec 26 2025: (Start)

Sum_{n>=0} 1/a(n) = 2^(-3/4) * BesselI(-3/4, 1) * Gamma(1/4).

Sum_{n>=0} (-1)^n/a(n) = 2^(-3/4) * BesselJ(-3/4, 1) * Gamma(1/4). (End)

G.f.: 3F0(1/4,1,1;;4*x). - R. J. Mathar, May 28 2026

Maple

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

Prog

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

Crossrefs

Cf. A006472, A001044, A000680, A068466, A084939, A084940, A084941, A084942, A084944, A085356.

Cf. A395757 (sum of reciprocals).

Sequence in context: A089906 A294515 A287317 * A055055 A222998 A368770

Adjacent sequences: A084940 A084941 A084942 * A084944 A084945 A084946

Keyword

easy,nonn

Author

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

Status

approved