A203312 - OEIS

%I #24 Nov 22 2023 09:44:06

%S 1,3,147,298116,47460365316,965460013501733568,

%T 3717096745012192786213464768,

%U 3763515081241454304168766426610670649344,1329626784930718063722475681347135527472012731205697536

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

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

%F a(n) ~ c * n^(n^2 - n - 2/3) / exp(3*n^2/2 - n*(n+1)*Pi/(2*sqrt(3)) - n), where c = Gamma(1/3) * 3^(1/12) * exp(Pi/(12*sqrt(3))) / (2^(4/3) * Pi^(4/3)) = 0.2945280196744096322469352538791946777977998766871923997662057483092872... - _Vaclav Kotesovec_, Nov 22 2023

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

%t v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,

%t {j, 1, k - 1}], {k, 2, n}]

%t Table[v[n], {n, 1, z}]          (* A203312 *)

%t Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)

%o (Python)

%o from operator import mul

%o from functools import reduce

%o def v(n): return 1 if n==1 else reduce(mul, [j**2 - j*k + k**2 for k in range(2, n + 1) for j in range(1, k)])

%o print([v(n) for n in range(1, 11)]) # _Indranil Ghosh_, Jul 26 2017

%Y Cf. A203012, A203673, A367543.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 04 2012