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A203312
Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).
7
1, 3, 147, 298116, 47460365316, 965460013501733568, 3717096745012192786213464768, 3763515081241454304168766426610670649344, 1329626784930718063722475681347135527472012731205697536
OFFSET
1,2
COMMENTS
See A093883 for a discussion and guide to related sequences.
FORMULA
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
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}] (* A203312 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)
PROG
(Python)
from operator import mul
from functools import reduce
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)])
print([v(n) for n in range(1, 11)]) # Indranil Ghosh, Jul 26 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 04 2012
STATUS
approved