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Search: id:A119440
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| A119440 |
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Triangle read by rows: T(n,k) is the number of binary sequences of length n that start with exactly k 01's (0<=k<=floor(n/2)). |
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+0
3
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| 1, 2, 3, 1, 6, 2, 12, 3, 1, 24, 6, 2, 48, 12, 3, 1, 96, 24, 6, 2, 192, 48, 12, 3, 1, 384, 96, 24, 6, 2, 768, 192, 48, 12, 3, 1, 1536, 384, 96, 24, 6, 2, 3072, 768, 192, 48, 12, 3, 1, 6144, 1536, 384, 96, 24, 6, 2, 12288, 3072, 768, 192, 48, 12, 3, 1, 24576, 6144, 1536, 384, 96
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n contains 1+floor(n/2) terms. Sum of entries in row n is 2^n (A000079). T(n,0)=A098011(n+2). Except for a shift, all columns are identical. G.f. of column k is x^(2k)*(1-x^2)/(1-2x). Sum(k*T(n,k),k=0..floor(n/2))=A000975(n-1).
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FORMULA
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T(n,k)=3*2^(n-2k-2) for n>=2k+2; T(2k,k)=1; T(2k+1,k)=2. G.f.=G(t,x)=(1-x^2)/[(1-2x)(1-tx^2)].
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EXAMPLE
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T(6,2)=3 because we have 010100, 010110 and 010111.
Triangle starts:
1;
2;
3,1;
6,2;
12,3,1;
24,6,2;
48,12,3,1;
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MAPLE
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T:=proc(n, k) if 2*k+2<=n then 3*2^(n-2*k-2) elif n=2*k then 1 elif n=2*k+1 then 2 else 0 fi end: for n from 0 to 16 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000079, A098011, A000975.
Sequence in context: A083855 A062565 A156344 this_sequence A165742 A162984 A166295
Adjacent sequences: A119437 A119438 A119439 this_sequence A119441 A119442 A119443
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 19 2006
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