OFFSET
0,2
COMMENTS
Note that a(n) = D(n,n,-2,0), where D(k,n,x,y) denotes the (n+1) X (n+1) determinant with (i,j)-entry equal to (i+j*x+y)^k for all i,j = 0,...,n. By the comments in A176113, it is known that D(n,n,x,y) = (-x)^{n*(n+1)/2}*(n!)^{n+1}. Note also that D(k,n,x,y) = 0 for all k = 0,...,n-1, which can be proved by using the definition of determinant and the binomial theorem.
For any matrices M of this pattern, M(i, j) = M(i-2, j-1). - Iain Fox, Feb 26 2018
REFERENCES
J. M. Monier, Algèbre et géometrie, Dunod, 1996.
LINKS
Iain Fox, Table of n, a(n) for n = 0..28
C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2017.
FORMULA
a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) as shown by comments. - Iain Fox, Apr 15 2018
EXAMPLE
Northwest corner of matrix corresponding to a(n):
0^n (-2)^n (-4)^n (-6)^n (-8)^n
1 (-1)^n (-3)^n (-5)^n (-7)^n
2^n 0 (-2)^n (-4)^n (-6)^n
3^n 1 (-1)^n (-3)^n (-5)^n
4^n 2^n 0 (-2)^n (-4)^n
MATHEMATICA
a[n_]:=Det[Table[If[n==0, 1, (i-2j)^n], {i, 0, n}, {j, 0, n}]]
Table[a[n], {n, 0, 10}]
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 19 2013
STATUS
approved