A105114 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 2.

1, 1, 1, 1, 2, 2, 4, 3, 1, 7, 6, 3, 12, 13, 6, 1, 21, 26, 13, 4, 37, 50, 30, 10, 1, 65, 96, 66, 24, 5, 114, 184, 139, 59, 15, 1, 200, 350, 288, 140, 40, 6, 351, 661, 591, 318, 105, 21, 1, 616, 1242, 1199, 704, 266, 62, 7, 1081, 2324, 2406, 1533, 645, 174, 28, 1, 1897, 4332

Offset: 0

Author

Emeric Deutsch, Apr 07 2005

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/2) terms. Row sums are the powers of 2 (A000079). Column 0 yields A005251.

Number of binary words of length n-1 having k isolated 0's. Example: T(5,1)=6 because we have 0111, 0100, 1011, 1101, 0010 and 1110. - Emeric Deutsch, May 21 2006

Examples

			T(7,3) = 4 because we have (1,2,2,2), (2,1,2,2), (2,2,1,2) and (2,2,2,1).
Triangle begins:
1;
1;
1,      1;
2,      2;
4,      3,    1;
7,      6,    3;
12,    13,    6,   1;
21,    26,   13,   4;
37,    50,   30,  10,   1;
65,    96,   66,  24,   5;
114,  184,  139,  59,  15,  1;
200,  350,  288, 140,  40,  6;
351,  661,  591, 318, 105, 21,  1;
616, 1242, 1199, 704, 266, 62,  7;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000079, A005251.

Programs

Maple

G:=(1-z)/(1-2*z-z^2*t+z^3*t+z^2-z^3):Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od;# yields sequence in triangular form

Mathematica

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[1/(1-(x/(1-x)-x^2+y x^2)),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 05 2012 *)

Formula

G.f.: (1-z)/(1-2z+z^2-z^3-tz^2+tz^3).