A279662 a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000

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

Table of n, a(n) for n=0..15.

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Index to sequences related to pyramidal numbers

Index entries for sequences related to factorial numbers

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

Adjacent sequences: A279659 A279660 A279661 * A279663 A279664 A279665

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 16 2016

Status

approved