A203012 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,...,n).

1, 7, 1729, 37616124, 135933424914924, 132432199651531695045312, 51437933151214684812682944045953088, 11056394929890243558409721156996503083526683082752, 1743892714865607005898689849291524734866677095031979100765833773056

Offset: 1

Author

Clark Kimberling, Jan 04 2012

Keywords

• nonn

Comments

See A093883 for a discussion and guide to related sequences.

Examples

			a(1)=1
a(2)=1^2+1*2+2^2=7
a(3)=(1^2+1*2+2^2)(1^3+1*3+3^2)(2^2+2*3+3^2)=1729.
		

Crossrefs

Cf. A203312, A203475, A203673, A367542.

Programs

Mathematica

f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 + f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203012 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203158 *)

Formula

a(n) ~ c * n^(n^2 - n - 5/6) * 3^(n*(3*n+1)/4) / exp(3*n^2/2 - n - n*(n+1)*Pi / (4*sqrt(3))), where c = sqrt(Gamma(1/3)) * 3^(5/24) * exp(Pi/(24*sqrt(3))) / (2^(7/6) * Pi^(7/6)) = 0.26001211479205772659823692637002123572622409280442625312217301129630097... - Vaclav Kotesovec, Nov 22 2023