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
Links
- Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
Crossrefs
Programs
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Maple
a := n->n!/2^n*mul(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];
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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]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 29 2019 *)
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PARI
a(n)=n!/2^n*prod(i=1,n,5*i-3) \\ Charles R Greathouse IV, Dec 13 2016
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