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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.

%I A228252 #27 Apr 15 2018 15:27:52
%S A228252 1,2,64,82944,8153726976,97844723712000000,
%T A228252 210357201231685877760000000,111759427954264225978066246041600000000,
%U A228252 19353724511515955943723861007628909886308352000000000,1393093075882582456065167957036969287436705021776979747143680000000000,51765823014530203817669442380756522498563227474168874049894256476160000000000000000000000
%N 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.
%C A228252 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.
%C A228252 For any matrices M of this pattern, M(i, j) = M(i-2, j-1). - _Iain Fox_, Feb 26 2018
%D A228252 J. M. Monier, Algèbre et géometrie, Dunod, 1996.
%H A228252 Iain Fox, <a href="/A228252/b228252.txt">Table of n, a(n) for n = 0..28</a>
%H A228252 C. Krattenthaler, <a href="https://arxiv.org/abs/math/0503507">Advanced Determinant Calculus: A Complement</a>, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2017.
%F A228252 a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) as shown by comments. - _Iain Fox_, Apr 15 2018
%e A228252 Northwest corner of matrix corresponding to a(n):
%e A228252 0^n  (-2)^n  (-4)^n  (-6)^n  (-8)^n
%e A228252   1  (-1)^n  (-3)^n  (-5)^n  (-7)^n
%e A228252 2^n       0  (-2)^n  (-4)^n  (-6)^n
%e A228252 3^n       1  (-1)^n  (-3)^n  (-5)^n
%e A228252 4^n     2^n       0  (-2)^n  (-4)^n
%t A228252 a[n_]:=Det[Table[If[n==0,1,(i-2j)^n],{i,0,n},{j,0,n}]]
%t A228252 Table[a[n],{n,0,10}]
%o A228252 (PARI) a(n) = matdet(matrix(n+1, n+1, i, j, (i - 2*j + 1)^n)) \\ _Iain Fox_, Feb 16 2018
%o A228252 (PARI) a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) \\ (faster and uses less memory) _Iain Fox_, Apr 15 2018
%Y A228252 Cf. A176113.
%K A228252 nonn
%O A228252 0,2
%A A228252 _Zhi-Wei Sun_, Aug 19 2013