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Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).
7

%I #15 Nov 25 2023 10:57:21

%S 1,21,254163,11213968422384,6451450005117349260375984,

%T 127857993263632065817610313129228311433216,

%U 191199773886534869435599958788731398661833328276349525268803584

%N Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

%C See A093883 for a discussion and guide to related sequences.

%F From _Vaclav Kotesovec_, Nov 22 2023: (Start)

%F a(n) = A203012(n) * A203312(n).

%F a(n) ~ c * 3^(n*(3*n+1)/4) * n^(2*n^2 - 2*n - 3/2) / exp(3*n^2 - n*(n+1)*Pi*sqrt(3)/4 - 2*n), where c = Gamma(1/3)^(3/2) * 3^(7/24) * exp(Pi/(8*sqrt(3))) / (2^(5/2) * Pi^(5/2)) = 0.076580853261060033865281896312127877504662138809362419847380161198324... (End)

%t f[j_] := j^2; z = 12;

%t u[n_] := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]

%t v[n_] := Product[u[n], {k, 2, n}]

%t Table[v[n], {n, 1, z}] (* A203673 *)

%t Table[v[n + 1]/v[n], {n, 1, z}] (* A203674 *)

%Y Cf. A367550.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 04 2012