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A101161
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A number triangle associated with the Chebyshev polynomials of the first kind.
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2
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1, 2, 1, 2, 3, 1, 2, 7, 4, 1, 2, 18, 14, 5, 1, 2, 47, 52, 23, 6, 1, 2, 123, 194, 110, 34, 7, 1, 2, 322, 724, 527, 198, 47, 8, 1, 2, 843, 2702, 2525, 1154, 322, 62, 9, 1, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 1, 2, 5778, 37634, 57965, 39202, 15127, 3842, 702, 98
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums are A101162. Diagonal sums are A101163.
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LINKS
| Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| Number triangle S(n, k)=if(n=k, 1, 2T(n-k, (k+2)/2)) where T(n, k)=(n/2)sum{j=0..floor(n/2), C(n-j, j)(-1)^j*(2k)^(n-2j)};or S(n, k)=if(k<n, sum{j=0..n, C(n-k+j, 2j)(2(n-k)/(n-k+j))k^j}, if(k=n, 1, 0)) Columns have g.f. (1-x^2)x^k/(1-(k+2)x+x^2). Also square array if(n=0, 1, 2T(n, (k+2)/2) read by antidiagonals.
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EXAMPLE
| Rows begin {1}, {2,1}, {2,3,1}, {2,7,4,1}, {2,18,14,5,1},...
As a square array, rows begin
1,1,1,1,1,...
2,3,4,5,6,...
2,7,14,23,34,...
2,18,52,110,198,...
2,47,194,527,1154,...
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CROSSREFS
| Sequence in context: A067763 A087730 A126247 * A097825 A002343 A082076
Adjacent sequences: A101158 A101159 A101160 * A101162 A101163 A101164
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Dec 02 2004
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