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A105114 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 2. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A252040 A084896 A011388 * A166284 A098086 A332887

Adjacent sequences:  A105111 A105112 A105113 * A105115 A105116 A105117

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Apr 07 2005

STATUS

approved

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Last modified March 30 06:59 EDT 2020. Contains 333119 sequences. (Running on oeis4.)