A084939 Pentagorials: n-th polygorial for k=5.

1, 1, 5, 60, 1320, 46200, 2356200, 164934000, 15173928000, 1775349576000, 257425688520000, 45306921179520000, 9514453447699200000, 2350070001581702400000, 674470090453948588800000, 222575129849803034304000000

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..243

M. A. Asiru, Sequence factorial of g-gonal numbers, Int. J. Math. Educ. Sci. Technol., 44(4) (2012), 579-586.

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

Formula

a(n) = polygorial(n, 5) = (A000142(n)/A000079(n))*A008544(n) = (n!/2^n)*Product_{i=0..n-1} (3*i+2) = (n!/2^n)*3^n*Pochhammer(2/3, n) = (n!/2^n)*3^n*GAMMA(n+2/3)/GAMMA(2/3).

a(n) ~ Gamma(1/3) * 3^(n+1/2) * n^(2*n+2/3) / (2^n * exp(2*n)). - Vaclav Kotesovec, Jul 17 2015

D-finite with recurrence a(n+1) = ((n+1)*(3*n+2)/2)*a(n) = A000326(n+1)*a(n). - Muniru A Asiru, Apr 05 2016

E.g.f.: hypergeom([2/3, 1], [], (3/2)*x). - Robert Israel, Apr 05 2016

Maple

a := n->(n!/2^n)*mul(3*i+2, i=0..n-1); [seq(a(j), j=0..30)];

Mathematica

Table[k! Pochhammer[2/3, k] (3/2)^k, {k, 0, 20}] (* Jan Mangaldan, Mar 20 2013 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[5, #] &, 17, 0] (* Robert G. Wilson v, Dec 17 2016 *)

Prog

(PARI) a(n)=n!/2^n*prod(i=1, n, 3*i-1) \\ Charles R Greathouse IV, Dec 13 2016

Crossrefs

Cf. A000326, A006472, A001044, A000680, A084940, A084941, A084942, A084943, A084944, A085356.

Sequence in context: A010793 A180614 A138447 * A171205 A214380 A218801

Adjacent sequences: A084936 A084937 A084938 * A084940 A084941 A084942

Keyword

easy,nonn

Author

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

Status

approved