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Search: id:A110135
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| A110135 |
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Square array of expansions of 1/sqrt(1-4x-4*k*x^2), read by anti-diagonals. |
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+0
3
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| 1, 2, 1, 6, 2, 1, 20, 8, 2, 1, 70, 32, 10, 2, 1, 252, 136, 44, 12, 2, 1, 924, 592, 214, 56, 14, 2, 1, 3432, 2624, 1052, 304, 68, 16, 2, 1, 12870, 11776, 5284, 1632, 406, 80, 18, 2, 1, 48620, 53344, 26840, 9024, 2332, 520, 92, 20, 2, 1, 184756, 243392, 137638, 50304
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Column k has g.f. 1/sqrt(1-4x-4*k*x^2) and e.g.f. exp(2x)BesselI(0,2*sqrt(k)x). Columns include A000984,A006139,A084609,A098453. Row sums of triangle are A110136. Diagonal sums of triangle are A110137.
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FORMULA
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Square array T(n, k)=sum{j=0..floor(n/2), C(n, j)C(2(n-j), n)k^j}; as a number triangle, T1(n, k)=sum{j=0..floor((n-k)/2), C(n-k, j)C(2(n-k-j), n-k)k^j};
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EXAMPLE
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As a square array, rows start
1,1,1,1,1,...
2,2,2,2,2,...
6,8,10,12,14,16,...
20,32,44,56,68,80,...
70,136,214,304,406,520,...
252,592,1052,1632,2332,3152,...
As a number triangle, rows start
1;
2,1;
6,2,1;
20,8,2,1;
70,30,10,2,1;
252,136,44,12,2,1;
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CROSSREFS
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Sequence in context: A124730 A114283 A106187 this_sequence A114423 A069123 A134133
Adjacent sequences: A110132 A110133 A110134 this_sequence A110136 A110137 A110138
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Jul 13 2005
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