A119440 Triangle read by rows: T(n,k) is the number of binary sequences of length n that start with exactly k 01's (0 <= k <= floor(n/2)).

1, 2, 3, 1, 6, 2, 12, 3, 1, 24, 6, 2, 48, 12, 3, 1, 96, 24, 6, 2, 192, 48, 12, 3, 1, 384, 96, 24, 6, 2, 768, 192, 48, 12, 3, 1, 1536, 384, 96, 24, 6, 2, 3072, 768, 192, 48, 12, 3, 1, 6144, 1536, 384, 96, 24, 6, 2, 12288, 3072, 768, 192, 48, 12, 3, 1, 24576, 6144, 1536, 384, 96

Offset

0,2

Comments

Row n contains 1+floor(n/2) terms.

Sum of entries in row n is 2^n (A000079).

T(n,0)=A098011(n+2). Except for a shift, all columns are identical.

G.f. of column k is x^(2k)*(1-x^2)/(1-2x).

Sum_{k=0..floor(n/2)} k*T(n,k) = A000975(n-1).

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

T(n,k) = 3*2^(n-2k-2) for n >= 2k+2; T(2k,k)=1; T(2k+1,k)=2.

G.f.: G(t,x) = (1-x^2)/((1-2*x)*(1-t*x^2)).

Example

T(6,2)=3 because we have 010100, 010110 and 010111.
Triangle starts:
   1;
   2;
   3,  1;
   6,  2;
  12,  3,  1;
  24,  6,  2;
  48, 12,  3,  1;

Maple

T:=proc(n, k) if 2*k+2<=n then 3*2^(n-2*k-2) elif n=2*k then 1 elif n=2*k+1 then 2 else 0 fi end: for n from 0 to 16 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form

Mathematica

nn=15; a=1/(1-y x^2); c=1/(1-2x); Map[Select[#, #>0&]&, CoefficientList[Series[1+x c+x^2 a c+x a +x^2y a+x^3y a c, {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Jan 03 2014 *)
CoefficientList[CoefficientList[Series[(1 - x^2)/((1 - 2*x)*(1 - y*x^2)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 10 2017 *)

Crossrefs

Cf. A000079, A000975, A098011.

Sequence in context: A175137 A156344 A218796 * A165742 A162984 A166295

Adjacent sequences: A119437 A119438 A119439 * A119441 A119442 A119443

Keyword

nonn,tabf

Author

Emeric Deutsch, May 19 2006

Status

approved