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A118390
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Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 000 (n,k>=0).
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2
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1, 2, 4, 7, 1, 13, 2, 1, 24, 5, 2, 1, 44, 12, 5, 2, 1, 81, 26, 13, 5, 2, 1, 149, 56, 29, 14, 5, 2, 1, 274, 118, 65, 32, 15, 5, 2, 1, 504, 244, 143, 74, 35, 16, 5, 2, 1, 927, 499, 307, 169, 83, 38, 17, 5, 2, 1, 1705, 1010, 652, 374, 196, 92, 41, 18, 5, 2, 1, 3136, 2027, 1369, 819
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row n has n-1 terms (n>=2). Sum of entries in row n is 2^n (A000079). T(n,0)=A000073(n+3) (the tribonacci numbers). T(n,1)=A073778(n-1). 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+(1-t)z^2]/[1-(1+t)z-(1-t)z^2-(1-t)z^3]. Recurrence relation: T(n,k)=T(n-1,k)+T(n-2,k)+T(n-3,k)+T(n-1,k-1)-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 000010,000011,010000,100001 and 110000.
Triangle starts:
1;
2;
4;
7,1;
13,2,1;
24,5,2,1;
44,12,5,2,1;
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MAPLE
| G:=(1+(1-t)*z+(1-t)*z^2)/(1-(1+t)*z-(1-t)*z^2-(1-t)*z^3): Gser:=simplify(series(G, z=0, 32)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: P[0]; P[1]; for n from 2 to 13 do seq(coeff(P[n], t, k), k=0..n-2) od; #yields sequence in triangular form
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CROSSREFS
| Cf. A000073, A073778, A001787, A000079, A076791.
Sequence in context: A118429 A110317 A098073 * A202848 A202841 A134974
Adjacent sequences: A118387 A118388 A118389 * A118391 A118392 A118393
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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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