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

0,5

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

Links

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

Formula

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

Example

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;

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 *)

Crossrefs

Cf. A000079, A005251.

Sequence in context: A084896 A011388 A349474 * A166284 A098086 A332887

Adjacent sequences: A105111 A105112 A105113 * A105115 A105116 A105117

Keyword

nonn,tabf

Author

Emeric Deutsch, Apr 07 2005

Status

approved