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A098593
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A triangle of Krawtchouk coefficients.
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5
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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; refs; listen; history; internal format)
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OFFSET
| 0,12
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COMMENTS
| 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
| P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks. Contemporary Mathematics, 287 2001, pp. 83-96
P. Feinsilver and J. Kocik, Krawtchouk polynomials and Krawtchouk matrices, quant-ph/0702073. [From Paul Barry (pbarry(AT)wit.ie), Oct 05 2010]
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FORMULA
| 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
T(n,k)=[k<=n]*Hypergeometric2F1(-k,k-n;1;-1). [From Paul Barry (pbarry(AT)wit.ie), 24 Jan 2011]
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EXAMPLE
| Rows begin {1}, {1,1}, {1,0,1}, {1,-1,-1,1}, {1,-2,-2,-2,1}, ...
Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 05 2010: (Start)
Triangle begins
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
Production matrix (related to large Schroeder numbers A006318) begins
1, 1,
0, -1, 1,
0, -2, -1, 1,
0, -6, -2, -1, 1,
0, -22, -6, -2, -1, 1,
0, -90, -22, -6, -2, -1, 1,
0, -394, -90, -22, -6, -2, -1, 1,
0, -1806, -394, -90, -22, -6, -2, -1, 1,
0, -8558, -1806, -394, -90, -22, -6, -2, -1, 1
Production matrix of inverse is
-1, 1,
-2, 1, 1,
-4, 2, 1, 1,
-8, 4, 2, 1, 1,
-16, 8, 4, 2, 1, 1,
-32, 16, 8, 4, 2, 1, 1,
-64, 32, 16, 8, 4, 2, 1, 1,
-128, 64, 32, 16, 8, 4, 2, 1, 1,
-256, 128, 64, 32, 16, 8, 4, 2, 1, 1 (End)
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CROSSREFS
| Cf. A081579.
Sequence in context: A172353 A104754 A206827 * A144431 A053821 A076545
Adjacent sequences: A098590 A098591 A098592 * A098594 A098595 A098596
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 17 2004
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