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A143361
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Triangle read by rows: T(n,k) is the number of 010-avoiding binary words of length n containing k 00 subwords (0<=k<=n-1).
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1
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2, 3, 1, 4, 2, 1, 6, 3, 2, 1, 9, 6, 3, 2, 1, 13, 11, 7, 3, 2, 1, 19, 18, 14, 8, 3, 2, 1, 28, 30, 24, 17, 9, 3, 2, 1, 41, 50, 43, 30, 20, 10, 3, 2, 1, 60, 81, 77, 57, 36, 23, 11, 3, 2, 1, 88, 130, 132, 108, 72, 42, 26, 12, 3, 2, 1, 129, 208, 224, 193, 143, 88, 48, 29, 13, 3, 2, 1
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OFFSET
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1,1
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COMMENTS
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Sum of entries in row n = A005251(n+3).
T(n,0) = A000930(n+2).
Sum(k*T(n,k), k=0..n-1) = A118430(n+1).
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LINKS
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Alois P. Heinz, Rows n = 1..150, flattened
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FORMULA
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G.f.: G(t,z) = (1+z-tz+z^2)/(1-z-tz+tz^2-z^3)-1.
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EXAMPLE
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T(5,2)=3 because we have 00011, 10001 and 11000.
Triangle starts:
2;
3, 1;
4, 2, 1;
6, 3, 2, 1;
9, 6, 3, 2, 1;
13, 11, 7, 3, 2, 1;
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MAPLE
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G:=(1+z-t*z+z^2)/(1-z-t*z+t*z^2-z^3)-1: Gser:=simplify(series(G, z=0, 14)): for n to 12 do P[n]:=sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j=0..n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<3,
expand(b(n-1, i+1) +b(n-1, i)*`if`(i=2, x, 1)), b(n-1, 1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..15); # Alois P. Heinz, Dec 18 2013
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<3, Expand[b[n-1, i+1] + b[n-1, i]*If[i == 2, x, 1]], b[n-1, 1]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A005251, A000930, A118430.
Sequence in context: A255250 A273620 A104705 * A152547 A083906 A361136
Adjacent sequences: A143358 A143359 A143360 * A143362 A143363 A143364
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Aug 15 2008
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STATUS
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approved
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