A228252 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to (i-2j)^n for all i,j = 0,...,n.

1, 2, 64, 82944, 8153726976, 97844723712000000, 210357201231685877760000000, 111759427954264225978066246041600000000, 19353724511515955943723861007628909886308352000000000, 1393093075882582456065167957036969287436705021776979747143680000000000, 51765823014530203817669442380756522498563227474168874049894256476160000000000000000000000

Offset: 0

Author

Zhi-Wei Sun, Aug 19 2013

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A176113.

Programs

Mathematica

a[n_]:=Det[Table[If[n==0,1,(i-2j)^n],{i,0,n},{j,0,n}]]
Table[a[n],{n,0,10}]

PARI

a(n) = matdet(matrix(n+1, n+1, i, j, (i - 2*j + 1)^n)) \\ Iain Fox, Feb 16 2018

PARI

a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) \\ (faster and uses less memory) Iain Fox, Apr 15 2018

Formula

a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) as shown by comments. - Iain Fox, Apr 15 2018