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A386650 Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.

%I A386650 #8 Aug 09 2025 11:53:01
%S A386650 1,1,3,30,930,93930,31560480,35600221440,136099646565120,
%T A386650 1776236487321381120,79580723341459838319360,
%U A386650 12296654209275691297430868480,6578267322410960919238807125534720,12223446894741861497849104893155093176320,79112201841847644246811045518121813092796661760
%N A386650 Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.
%C A386650 Conjecture: 2*a(n) = A214590(n) - 2 for n >= 1, where A214590(n) is the number of nXnXn triangular 0..3 arrays with every horizontal row having the same average value.
%H A386650 Paul D. Hanna, <a href="/A386650/b386650.txt">Table of n, a(n) for n = 0..70</a>
%F A386650 a(n) = Product_{k=0..n} A005725(k) for n >= 0.
%F A386650 a(n) ~ c * exp(n/2) * (11 + 217/(6371 + 624*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^(-1 + n/2 + n^2/2) * ((39 + (4563 - 78*sqrt(78))^(1/3) + (4563 + 78*sqrt(78))^(1/3))/13)^(n/2) / (2^(-11/4 + 2*n + n^2) * 3^((-3 + 2*n + n^2)/2) * Pi^(n/2 + 1/4) * n^((4290 - 1421*78^(2/3)/(804726 - 73709*sqrt(78))^(1/3) - (78*(804726 - 73709*sqrt(78)))^(1/3) + 4056*n)/8112)), where c = 0.77060824350557924602665408964165291884080801923663... - _Vaclav Kotesovec_, Aug 09 2025
%e A386650 The quadrinomial coefficients A005725(n) = [x^n] (1 + x + x^2 + x^3)^n for n >= 0 begin [1, 1, 3, 10, 31, 101, 336, 1128, 3823, ...], where a(n) equals the product of the terms A005725(0) through A005725(n).
%t A386650 Table[Product[HypergeometricPFQ[{(1-k)/2, -k, -k/2}, {1/2, 1}, -1], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Aug 09 2025 *)
%o A386650 (PARI) {a(n) = prod(k=0, n, polcoef((1 + x + x^2 + x^3)^k, k) )}
%o A386650 for(n=0, 15, print1(a(n), ", "))
%Y A386650 Cf. A214590, A386649, A342177, A003046, A005725.
%K A386650 nonn
%O A386650 0,3
%A A386650 _Paul D. Hanna_, Aug 08 2025