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A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.

%I A386649 #23 Aug 11 2025 10:31:18
%S A386649 1,1,3,21,399,20349,2869209,1127599137,1248252244659,3918263795984601,
%T A386649 35080215765450132753,899912775031092255512709,
%U A386649 66403663756769266442027284401,14140062564030204365431731967633341,8713488333644640745496899895218790824407
%N A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.
%C A386649 Conjecture: a(n) = A214589(n) - 2 for n >= 1, where A214589(n) is the number of n X n X n triangular 0..2 arrays with every horizontal row having the same average value.
%H A386649 Paul D. Hanna, <a href="/A386649/b386649.txt">Table of n, a(n) for n = 0..70</a>
%F A386649 a(n) = Product_{k=0..n} A002426(k) for n >= 0.
%F A386649 a(n) ~ c * 3^((n-1)*(n+3)/2) * exp(n/2) / (2^(n - 3/4) * Pi^(n/2 - 1/4) * n^(n/2 + 7/16)), where c = 1.123782729130753266489882099159237662230713685736... - _Vaclav Kotesovec_, Aug 09 2025
%e A386649 The central trinomial coefficients A002426(n) = [x^n] (1 + x + x^2)^n for n >= 0 begin [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, ...], where a(n) equals the product of the terms A002426(0) through A002426(n).
%t A386649 Table[Product[3^k * Hypergeometric2F1[1/2, -k, 1, 4/3], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Aug 09 2025 *)
%o A386649 (PARI) {a(n) = prod(k=0,n, polcoef((1 + x + x^2)^k, k) )}
%o A386649 for(n=0,15,print1(a(n),", "))
%Y A386649 Cf. A214589, A386650, A342177, A003046, A002426.
%K A386649 nonn
%O A386649 0,3
%A A386649 _Paul D. Hanna_, Aug 08 2025