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%I #8 Sep 08 2022 08:46:18
%S 1,1,8,208,12480,1435200,281299200,86640153600,39507910041600,
%T 25482601976832000,22424689739612160000,26147188236387778560000,
%U 39429959860472770068480000,75350653293363463600865280000,179334554838205043370059366400000,523656900127558726640573349888000000
%N a(n) = (5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5).
%C Heptagonal pyramidal factorial numbers.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalPyramidalNumber.html">Heptagonal 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)*(5*k - 2)/6, a(0)=1.
%F a(n) = Product_{k=1..n} A002413(k), a(0)=1.
%F a(n) ~ (2*Pi)^(3/2)*(5/6)^n*n^(3*n+21/10)/(Gamma(3/5)*exp(3*n)).
%t FullSimplify[Table[(5/6)^n Gamma[n + 3/5] Gamma[n + 1] Gamma[n + 2]/Gamma[3/5], {n, 0, 15}]]
%o (Magma) [Round((5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5)): n in [0..20]]; // _Vincenzo Librandi_ Dec 17 2016
%Y Cf. A002413.
%Y Cf. A084940 (heptagonal factorial numbers).
%Y Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279662 (hexagonal pyramidal factorial numbers).
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 16 2016
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