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A279662
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a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).
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1
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1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000
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OFFSET
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0,3
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COMMENTS
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Hexagonal pyramidal factorial numbers.
More generally, the m-gonal pyramidal factorial numbers is 6^(-n)*(m-2)^n*Gamma(n+1)*Gamma(n+2)*Gamma(n+3/(m-2))/Gamma(3/(m-2)), m>2.
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} k*(k + 1)*(4*k - 1)/6, a(0)=1.
a(n) = Product_{k=1..n} A002412(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(2/3)^n*n^(3*n+9/4)/(Gamma(3/4)*exp(3*n)).
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MATHEMATICA
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FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]
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PROG
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(Magma) [Round((2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2) / Gamma(3/4)): n in [0..20]]; // Vincenzo Librandi, Dec 17 2016
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CROSSREFS
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Cf. A000680 (hexagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279663 (heptagonal pyramidal factorial numbers).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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