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%I #8 Sep 08 2022 08:46:18
%S 1,1,7,154,7700,731500,117771500,29678418000,11040371496000,
%T 5796195035400000,4144279450311000000,3920488359994206000000,
%U 4790836775912919732000000,7411424492337286825404000000,14266992147749277138902700000000,33670101468688294047810372000000000
%N a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).
%C Hexagonal pyramidal factorial numbers.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = Product_{k=1..n} k*(k + 1)*(4*k - 1)/6, a(0)=1.
%F a(n) = Product_{k=1..n} A002412(k), a(0)=1.
%F a(n) ~ (2*Pi)^(3/2)*(2/3)^n*n^(3*n+9/4)/(Gamma(3/4)*exp(3*n)).
%t FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]
%o (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
%Y Cf. A002412.
%Y Cf. A000680 (hexagonal factorial numbers).
%Y Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279663 (heptagonal pyramidal factorial numbers).
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 16 2016
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