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

%I A203012 #15 Nov 22 2023 09:42:55
%S A203012 1,7,1729,37616124,135933424914924,132432199651531695045312,
%T A203012 51437933151214684812682944045953088,
%U A203012 11056394929890243558409721156996503083526683082752,1743892714865607005898689849291524734866677095031979100765833773056
%N A203012 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,...,n).
%C A203012 See A093883 for a discussion and guide to related sequences.
%F A203012 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
%e A203012 a(1)=1
%e A203012 a(2)=1^2+1*2+2^2=7
%e A203012 a(3)=(1^2+1*2+2^2)(1^3+1*3+3^2)(2^2+2*3+3^2)=1729.
%t A203012 f[j_] := j; z = 12;
%t A203012 v[n_] := Product[Product[f[j]^2 + f[j] f[k] + f[k]^2,
%t A203012 {j, 1, k - 1}], {k, 2, n}]
%t A203012 Table[v[n], {n, 1, z}]          (* A203012 *)
%t A203012 Table[v[n + 1]/v[n], {n, 1, z}] (* A203158 *)
%Y A203012 Cf. A203312, A203475, A203673, A367542.
%K A203012 nonn
%O A203012 1,2
%A A203012 _Clark Kimberling_, Jan 04 2012