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A143291
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Triangle T(n,k), n>=2, 0<=k<=n-2, read by rows: numbers of binary words of length n containing at least one subword 10^{k}1 and no subwords 10^{i}1 with i<k.
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13
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1, 3, 1, 8, 2, 1, 19, 4, 2, 1, 43, 8, 3, 2, 1, 94, 15, 5, 3, 2, 1, 201, 27, 9, 4, 3, 2, 1, 423, 48, 15, 6, 4, 3, 2, 1, 880, 84, 24, 10, 5, 4, 3, 2, 1, 1815, 145, 38, 16, 7, 5, 4, 3, 2, 1, 3719, 248, 60, 24, 11, 6, 5, 4, 3, 2, 1, 7582, 421, 94, 35, 17, 8, 6, 5, 4, 3, 2, 1, 15397, 710, 146, 51, 25, 12, 7, 6, 5, 4, 3, 2, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 2..10012
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FORMULA
| G.f. of column k: x^(k+2) / ((x^(k+1)+x-1)*(x^(k+2)+x-1)).
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EXAMPLE
| T (5,1) = 4, because there are 4 words of length 5 containing at least one subword 101 and no subword 11: 00101, 01010, 10100, 10101.
Triangle begins:
1
3, 1
8, 2, 1
19, 4, 2, 1
43, 8, 3, 2, 1
94, 15, 5, 3, 2, 1
201, 27, 9, 4, 3, 2, 1
423, 48, 15, 6, 4, 3, 2, 1
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MAPLE
| as:= proc (n, k) option remember;
if k=0 then 2^n
elif n<=k and n>=0 then n+1
elif n>0 then as(n-1, k) +as(n-k-1, k)
else as(n+1+k, k) -as(n+k, k)
fi
end:
T:= (n, k)-> as(n, k) -as(n, k+1):
seq (seq (T(n, k), k=0..n-2), n=2..15);
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CROSSREFS
| Columns k=0-10 give: A008466, A143281, A143282, A143283, A143284, A143285, A143286, A143287, A143288, A143289, A143290.
Row sums are in A000295.
Sequence in context: A179393 A112420 A010288 * A200064 A077111 A073072
Adjacent sequences: A143288 A143289 A143290 * A143292 A143293 A143294
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KEYWORD
| nonn,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2008
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