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A118429
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Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 010 (n,k>=0).
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2
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1, 2, 4, 7, 1, 12, 4, 21, 10, 1, 37, 22, 5, 65, 47, 15, 1, 114, 98, 38, 6, 200, 199, 91, 21, 1, 351, 396, 210, 60, 7, 616, 777, 468, 158, 28, 1, 1081, 1508, 1014, 396, 89, 8, 1897, 2900, 2151, 952, 255, 36, 1, 3329, 5534, 4487, 2212, 687, 126, 9, 5842, 10492, 9229
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row n has ceil(n/2) terms (n>=1). Sum of entries in row n is 2^n (A000079). T(n,0)=A005251(n+3) T(n,1)=A118430(n). Sum(k*T(n,k),k=0..n-1)=(n-2)*2^(n-3) (A001787).
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FORMULA
| G.f.=G(t,z)=[1+(1-t)z^2]/[1-2z+(1-t)z^2-(1-t)z^3]. Recurrence relation: T(n,k)=2T(n-1,k)-T(n-2,k)+T(n-3,k)+T(n-2,k-1)-T(n-3,k-1) for n>=3.
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EXAMPLE
| T(6,2)=5 because we have 010010,010100,010101,001010 and 101010.
Triangle starts:
1;
2;
4;
7,1;
12,4;
21,10,1;
37,22,5;
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MAPLE
| G:=(1+(1-t)*z^2)/(1-2*z+(1-t)*z^2-(1-t)*z^3): Gser:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 16 do P[n]:=sort(coeff(Gser, z^n)) od: 1; for n from 1 to 16 do seq(coeff(P[n], t, j), j=0..ceil(n/2)-1) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000079, A005251, A001787, A118430.
Sequence in context: A061501 A089349 A118424 * A110317 A098073 A118390
Adjacent sequences: A118426 A118427 A118428 * A118430 A118431 A118432
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2006
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