A084940 Heptagorials: n-th polygorial for k=7.

1, 1, 7, 126, 4284, 235620, 19085220, 2137544640, 316356606720, 59791398670080, 14050978687468800, 4018579904616076800, 1374354327378698265600, 553864793933615401036800, 259762588354865623086259200, 140271797711627436466579968000, 86407427390362500863413260288000

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, 7) = (A000142(n)/A000079(n))*A047055(n) = (n!/2^n)*Product_{i=0..n-1}(5*i+2) = (n!/2^n)*5^n*Pochhammer(2/5, n) = (n!/2^n)*5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi.

D-finite with recurrence 2*a(n) = n*(5*n-3)*a(n-1). - R. J. Mathar, Mar 12 2019

a(n) ~ 5^n * n^(2*n + 2/5) * Pi /(Gamma(2/5) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

From Amiram Eldar, Dec 26 2025: (Start)

Sum_{n>=0} 1/a(n) = (2/5)^(3/10) * BesselI(-3/5, 2*sqrt(2/5)) * Gamma(2/5).

Sum_{n>=0} (-1)^n/a(n) = (2/5)^(3/10) * BesselJ(-3/5, 2*sqrt(2/5)) * Gamma(2/5). (End)

Maple

a := n->n!/2^n*mul(5*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[7, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Join[{1}, FoldList[Times, PolygonalNumber[7, Range[20]]]] (* Harvey P. Dale, Jul 29 2019 *)

Prog

(PARI) a(n)=n!/2^n*prod(i=1, n, 5*i-3) \\ Charles R Greathouse IV, Dec 13 2016

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084941, A084942, A084943, A084944, A085356, A246745.

Cf. A395538 (sum of reciprocals).

Sequence in context: A295412 A202798 A204248 * A246648 A308378 A139987

Adjacent sequences: A084937 A084938 A084939 * A084941 A084942 A084943

Keyword

easy,nonn

Author

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

Status

approved