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A111394 a(n) = product of first n integers not divisible by 3.

%I A111394 #19 Feb 28 2023 14:55:29
%S A111394 1,2,8,40,280,2240,22400,246400,3203200,44844800,717516800,
%T A111394 12197785600,231757926400,4635158528000,101973487616000,
%U A111394 2345390215168000,58634755379200000,1524503639859200000,42686101916057600000,1237896955565670400000,38374805622535782400000
%N A111394 a(n) = product of first n integers not divisible by 3.
%H A111394 Alois P. Heinz, <a href="/A111394/b111394.txt">Table of n, a(n) for n = 1..160</a>
%F A111394 Let b(n) = n!/(3^floor(n/3)*floor(n/3)!) then a(n) = b(3*floor(n/2)+(-1)^(n+1)). # formula corrected _Peter Luschny_, Oct 03 2012
%F A111394 E.g.f.: 1/(1 - x*G(x^2/3)) - 1 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764. - _Paul D. Hanna_, Jan 04 2014
%F A111394 E.g.f. A(x) satisfies: A'(x) = (1 + A(x))^3 * (1 + A(-x)). - _Paul D. Hanna_, Jan 04 2014
%e A111394 a(5) = 1*2*4*5*7 = 280.
%p A111394 a:= proc(n) a(n):= `if`(n=1, 1, (3*iquo(n, 2)-(-1)^n)*a(n-1)) end:
%p A111394 seq (a(n), n=1..25);  # _Alois P. Heinz_, Oct 03 2012
%t A111394 a[n_] := ((3*n-Mod[n, 2])/2)!/(3^((n-Mod[n, 2])/2)*((n-Mod[n, 2])/2)!); Table[a[n], {n, 1, 21}] (* _Jean-François Alcover_, Feb 25 2014 *)
%t A111394 FoldList[Times,Table[If[Divisible[n,3],Nothing,n],{n,40}]] (* _Harvey P. Dale_, Feb 28 2023 *)
%o A111394 (Sage)
%o A111394 def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
%o A111394 def A111394_list(n): return sorted(set([Gauss_factorial(j, 3) for j in (1..n)]))
%o A111394 A111394_list(28) # _Peter Luschny_, Oct 01 2012
%o A111394 (PARI) {a(n)=local(A,G=sum(k=0,n,binomial(3*k,k)/(2*k+1)*x^k +x*O(x^n))); A=1/(1-x*subst(G,x,x^2/3))-1; n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Jan 04 2014
%o A111394 (PARI) {a(n)=local(A=x); for(i=0, n, A=intformal((1+A)^3*subst(1+A, x,-x +x*O(x^n))^1 +x*O(x^n) )); n!*polcoeff(A, n)} \\ _Paul D. Hanna_, Jan 04 2014
%K A111394 nonn
%O A111394 1,2
%A A111394 _Jon Perry_, Nov 11 2005