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 A276482 a(n) = 5^n*Gamma(n+1/5)*Gamma(n+1)/Gamma(1/5). 1
 1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12). More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2. LINKS Robert Israel, Table of n, a(n) for n = 0..220 Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003. Index entries for sequences related to factorial numbers FORMULA a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1. a(n) = Product_{k=1..n} A051624(k), a(0)=1. a(n) = A000142(n)*A008548(n). a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)). Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914... MAPLE seq(mul(k*(5*k-4), k=1..n), n=0..20); # Robert Israel, Sep 18 2016 MATHEMATICA FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]] polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[12, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *) PROG (PARI) a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016 CROSSREFS Cf. A000142, A008548, A051624. Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11). Sequence in context: A308129 A356258 A286038 * A202788 A285028 A292784 Adjacent sequences: A276479 A276480 A276481 * A276483 A276484 A276485 KEYWORD nonn,easy AUTHOR Ilya Gutkovskiy, Sep 05 2016 STATUS approved

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Last modified September 29 18:32 EDT 2023. Contains 365775 sequences. (Running on oeis4.)