A203673 Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

1, 21, 254163, 11213968422384, 6451450005117349260375984, 127857993263632065817610313129228311433216, 191199773886534869435599958788731398661833328276349525268803584

Offset: 1

Author

Clark Kimberling, Jan 04 2012

Keywords

• nonn

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A367550.

Programs

Mathematica

f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203673 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203674 *)

Formula

From Vaclav Kotesovec, Nov 22 2023: (Start)

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

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)