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Search: id:A098593
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| A098593 |
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A triangle of Krawtchouk coefficients. |
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+0
4
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| 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -2, 1, 1, -3, -2, -2, -3, 1, 1, -4, -1, 0, -1, -4, 1, 1, -5, 1, 3, 3, 1, -5, 1, 1, -6, 4, 6, 6, 6, 4, -6, 1, 1, -7, 8, 8, 6, 6, 8, 8, -7, 1, 1, -8, 13, 8, 2, 0, 2, 8, 13, -8, 1, 1, -9, 19, 5, -6, -10, -10, -6, 5, 19, -9, 1, 1, -10, 26, -2, -17, -20, -20, -20, -17, -2, 26, -10, 1, 1, -11, 34, -14, -29, -25
(list; table; graph; listen)
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OFFSET
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0,12
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COMMENT
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Row sums are A009545(n+1), with e.g.f. exp(x)(cos(x)+sin(x)). Diagonal sums are A077948.
The rows are the diagonals of the Krawtchouk matrices. Coincides with the Riordan array (1/(1-x),(1-2x)/(1-x)). - Paul Barry (pbarry(AT)wit.ie), Sep 24 2004
Corresponds to Pascal-(1,-2,1) array, read by anti-diagonals. The Pascal-(1,-2,1) array has n-th row generated by (1-2x)^n/(1-x)^(n+1). - Paul Barry (pbarry(AT)wit.ie), Sep 24 2004
A modified version (different signs) of this triangle is given by T(n,k)=sum{j=0..n, C(n-k,j)C(k,j)cos(pi*(k-j))}; - Paul Barry (pbarry(AT)wit.ie), Jun 14 2007
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REFERENCES
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P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks. Contemporary Mathematics, 287 2001, pp. 83-96
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FORMULA
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Triangle T(n, k)=sum{i=0..k, binomial(n-k, k-i)binomial(k, i)(-1)^(k-i)}, k<=n.
T(n, k)=T(n-1, k)+T(n-1, k-1)-2T(n-2, k-1) (n>0). - Paul Barry (pbarry(AT)wit.ie), Sep 24 2004
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EXAMPLE
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Rows begin {1}, {1,1}, {1,0,1}, {1,-1,-1,1}, {1,-2,-2,-2,1}, ...
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CROSSREFS
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Cf. A081579.
Sequence in context: A078826 A051950 A104754 this_sequence A053821 A076545 A118400
Adjacent sequences: A098590 A098591 A098592 this_sequence A098594 A098595 A098596
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KEYWORD
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easy,sign,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Sep 17 2004
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