A109435 Triangle read by rows: T(n,m) = number of binary numbers n digits long, which have m 0's as a substring.

1, 2, 1, 4, 3, 1, 8, 7, 3, 1, 16, 15, 8, 3, 1, 32, 31, 19, 8, 3, 1, 64, 63, 43, 20, 8, 3, 1, 128, 127, 94, 47, 20, 8, 3, 1, 256, 255, 201, 107, 48, 20, 8, 3, 1, 512, 511, 423, 238, 111, 48, 20, 8, 3, 1, 1024, 1023, 880, 520, 251, 112, 48, 20, 8, 3, 1, 2048, 2047, 1815, 1121, 558

Offset: 0

Author

Robert G. Wilson v, Jun 28 2005

Keywords

• base
• nonn
• tabl

Comments

Column 0 is A000079, column 2 is A000225, column 3 is A008466, column 4 is A050231

Column 5 is A050232, column 6 is A050233, the last column is A001792.

A050227 with a leading column of powers of 2. - R. J. Mathar, Mar 25 2014

Examples

			Triangle begins:
n\m_0__1__2__3__4__5
0|  1  0  0  0  0  0
1|  2  1  0  0  0  0
2|  4  3  1  0  0  0
3|  8  7  3  1  0  0
4| 16 15  8  3  1  0
5| 32 31 19  8  3  1
T(5,3)=8 because there are 8 length 5 binary words that contain 000 as a contiguous substring:  00000, 00001, 00010, 00011, 01000, 10000, 10001, 11000. - _Geoffrey Critzer_, Jan 07 2014
		

Crossrefs

Cf. A109433, A001792, A109436, A102712 (row sums ?).

Programs

Maple

A109435 := proc(n,k)
    option remember ;
    if n< k then
        0;
    elif n = k then
        1;
    else
        2*procname(n-1,k)+2^(n-1-k)-procname(n-1-k,k) ;
    end if;
end proc:
seq(seq( A109435(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 05 2025

Mathematica

T[n_, m_] := Length[ Select[ StringPosition[ #, StringDrop[ ToString[10^m], 1]] & /@ Table[ ToString[ FromDigits[ IntegerDigits[i, 2]]], {i, 2^n, 2^(n + 1) - 1}], # != {} &]]; Flatten[ Table[ T[n, m], {n, 0, 11}, {m, 0, n}]]
nn=15;Map[Select[#,#>0&]&,Transpose[Table[CoefficientList[Series[x^m/(1-Sum[x^k,{k,1,m}])/(1-2x),{x,0,nn}],x],{m,0,nn}]]]//Grid (* Geoffrey Critzer, Jan 07 2014 *)

Formula

G.f. for column m: x^m/( (1 - Sum_{k=1..m} x^k)*(1-2*x) ). - Geoffrey Critzer, Jan 07 2014