A118429 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 010 (n,k >= 0).

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

Offset: 0

Author

Emeric Deutsch, Apr 27 2006

Keywords

• nonn
• tabf

Comments

Row n has ceiling(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=0..n-1} k*T(n,k) = (n-2)*2^(n-3) (A001787).

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..199, flattened

Crossrefs

Cf. A000079, A005251, A001787, A118430.

Programs

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

Mathematica

nn=15;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2z-(u-1)z^3/(1-(u-1)z^2)),{z,0,nn}],{z,u}]]//Grid (* Geoffrey Critzer, Dec 03 2013 *)

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.