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Search: id:A091917
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| A091917 |
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Coefficient array of polynomials (z-1)^n-1. |
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+0
2
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| 1, -2, 1, 0, -2, 1, -2, 3, -3, 1, 0, -4, 6, -4, 1, -2, 5, -10, 10, -5, 1, 0, -6, 15, -20, 15, -6, 1, -2, 7, -21, 35, -35, 21, -7, 1, 0, -8, 28, -56, 70, -56, 28, -8, 1, -2, 9, -36, 84, -126, 126, -84, 36, -9, 1, 0, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 11, -55, 165, -330, 462, -462, 330, -165
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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The first element has been changed to 1 to produce an invertible matrix. Alternatively, this is the coefficient array for the polynomials P(z,n)= product{j=0..n-1, z-(1+w(n)^j)} where w(n)=e^(2pi*i/n), i=sqrt(-1).
The row entries determine interesting recurrences. For instance, a(n)=4a(n-1)+6a(n-2)+4a(n-3), a(0)=a(1)=a(2)=1, gives A038503. Sequences of the form a(n)=sum{k=0..n, if (mod(k,m)=r, binomial(n,k), 0)}, for r=0..m-1, result. Equivalently, a(n)=sum{j=0..n-1, 2^n(cos(pi*j/m))^n*cos((n-2r)pi*j/m)}/m, r=0..m-1. These include A024493, A024494, A024495, A038503, A038504, A038505. The inverse matrix is A091918.
Triangle T(n,k), 0<=k<=n, read by rows given by [ -2, 2, 1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 11 2007
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EXAMPLE
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Rows begin {1}, {-2,1}, {0,-2,1}, {-2, 3, -3, 1}, {0,-4, 6, -4, 1},...
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CROSSREFS
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Sequence in context: A130210 A035443 A036261 this_sequence A025657 A025686 A022329
Adjacent sequences: A091914 A091915 A091916 this_sequence A091918 A091919 A091920
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KEYWORD
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sign,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Feb 13 2004
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