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A228252
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Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to (i-2j)^n for all i,j = 0,...,n.
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1
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1, 2, 64, 82944, 8153726976, 97844723712000000, 210357201231685877760000000, 111759427954264225978066246041600000000, 19353724511515955943723861007628909886308352000000000, 1393093075882582456065167957036969287436705021776979747143680000000000, 51765823014530203817669442380756522498563227474168874049894256476160000000000000000000000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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J. M. Monier, Algèbre et géometrie, Dunod, 1996.
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LINKS
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FORMULA
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a(n) = 2^(n*(n+1)/2)*(n!)^(n+1) as shown by comments. - Iain Fox, Apr 15 2018
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EXAMPLE
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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
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MATHEMATICA
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a[n_]:=Det[Table[If[n==0, 1, (i-2j)^n], {i, 0, n}, {j, 0, n}]]
Table[a[n], {n, 0, 10}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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