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A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

%I A290000 #57 Sep 17 2024 11:02:41
%S A290000 1,1,4,40,1120,91840,22408960,16358540800,35792487270400,
%T A290000 234870301468364800,4623187014103292723200,
%U A290000 272999193182799435304960000,48361261073946554365403054080000,25701205307660304745058529866383360000,40976048450930207702360695570691784048640000
%N A290000 a(n) = Product_{k=1..n-1} (3^k + 1).
%H A290000 G. C. Greubel, <a href="/A290000/b290000.txt">Table of n, a(n) for n = 0..50</a>
%F A290000 G.f. A(x) satisfies: A(x) = 1 + x * A(3*x) / (1 - x).
%F A290000 G.f.: Sum_{k>=0} 3^(k*(k - 1)/2) * x^k / Product_{j=0..k-1} (1 - 3^j*x).
%F A290000 a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * a(k).
%F A290000 a(n) ~ c * 3^(n*(n - 1)/2), where c = Product_{k>=1} (1 + 1/3^k) = 1.564934018567011537938849... = A132324.
%F A290000 a(n) = 3^(binomial(n+1,2))*(-1/3;1/3)_{n}, where (a;q)_{n} is the q-Pochhammer symbol. - _G. C. Greubel_, Feb 21 2021
%t A290000 Table[Product[3^k + 1, {k, 1, n - 1}], {n, 0, 14}]
%o A290000 (PARI) a(n) = prod(k=1, n-1, 3^k + 1); \\ _Michel Marcus_, Jun 06 2020
%o A290000 (Sage)
%o A290000 from sage.combinat.q_analogues import q_pochhammer
%o A290000 [1]+[3^(binomial(n,2))*q_pochhammer(n-1, -1/3, 1/3) for n in (1..20)] # _G. C. Greubel_, Feb 21 2021
%o A290000 (Magma)
%o A290000 [n lt 3 select 1 else (&*[3^j +1: j in [1..n-1]]): n in [1..20]]; // _G. C. Greubel_, Feb 21 2021
%Y A290000 Cf. A027871, A034472, A047656, A119600, A132324, A323716.
%Y A290000 Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), this sequence (m=3), A309327 (m=4).
%K A290000 nonn
%O A290000 0,3
%A A290000 _Ilya Gutkovskiy_, Jun 06 2020