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A279662 a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4). 1
1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
FORMULA
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)).
MATHEMATICA
FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]
PROG
(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
CROSSREFS
Cf. A002412.
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).
Sequence in context: A144683 A229711 A296232 * A214382 A141835 A111831
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 16 2016
STATUS
approved

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Last modified March 28 14:37 EDT 2024. Contains 371254 sequences. (Running on oeis4.)