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%I #36 May 02 2022 08:02:32
%S 1,1,7,126,4284,235620,19085220,2137544640,316356606720,
%T 59791398670080,14050978687468800,4018579904616076800,
%U 1374354327378698265600,553864793933615401036800,259762588354865623086259200
%N Heptagorials: n-th polygorial for k=7.
%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.
%F 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.
%F D-finite with recurrence 2*a(n) = n*(5*n-3)*a(n-1). - _R. J. Mathar_, Mar 12 2019
%p a := n->n!/2^n*mul(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];
%t 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 *)
%t Join[{1},FoldList[Times,PolygonalNumber[7,Range[20]]]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 29 2019 *)
%o (PARI) a(n)=n!/2^n*prod(i=1,n,5*i-3) \\ _Charles R Greathouse IV_, Dec 13 2016
%Y Cf. A006472, A001044, A000680, A084939, A084941, A084942, A084943, A084944, A085356.
%K easy,nonn
%O 0,3
%A Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
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