A282714 Base-2 generalized Pascal triangle P_2 read by rows (see Comments for precise definition).

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 1, 3, 0, 3, 0, 0, 0, 1, 1, 1, 3, 0, 3, 0, 0, 0, 1, 1, 2, 2, 1, 1, 2, 0, 0, 0, 1, 1, 2, 3, 1, 1, 1, 1, 0, 0, 0, 1, 1, 3, 1, 3, 0, 2, 0, 1, 0, 0, 0, 1, 1, 2, 4, 1, 2, 0, 2, 0, 0

Offset: 0

Author

N. J. A. Sloane, Mar 02 2017

Keywords

• nonn
• tabl

Comments

List the binary numbers in their natural order as binary strings, beginning with the empty string epsilon, which represents 0. Row n of the triangle gives the number of times the k-th string occurs as a (scattered) substring of the n-th string.

Row n has sum n+1.

Examples

			Triangle begins:
  1,
  1,1,
  1,1,1,
  1,2,0,1,
  1,1,2,0,1,
  1,2,1,1,0,1,
  1,2,2,1,0,0,1,
  1,3,0,3,0,0,0,1,
  1,1,3,0,3,0,0,0,1
  1,2,2,1,1,2,0,0,0,1
  1,2,3,1,1,1,1,0,0,0,1
  1,3,1,3,0,2,0,1,0,0,0,1
  1,2,4,1,2,0,2,0,0,0,0,0,1
  ...
The binary numbers are epsilon, 1, 10, 11, 100, 101, 110, 111, 1000, ...
The fifth number 101 contains
eps 1 10 11 100 101 respectively
.1..2..1..1...0...1 times, which is row 5 of the triangle.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 0..10000
• Julien Leroy, Michel Rigo, Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881.
• Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
• Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
• Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

Crossrefs

A007306 gives (essentially) the number of nonzero entries in the rows.

Programs

Maple

Nscatsub := proc(subw,w)
    local lsubw,lw,N,wri,wr,i ;
    lsubw := nops(subw) ;
    lw := nops(w) ;
    N := 0 ;
    if lsubw = 0 then
        return 1 ;
    elif lsubw > lw then
        return 0 ;
    else
        for wri in combinat[choose](lw,lsubw) do
            wr := [] ;
            for i in wri do
                wr := [op(wr),op(i,w)] ;
            end do:
            if verify(subw,wr,'sublist') then
                N := N+1 ;
            end if;
        end do:
    end if;
    return N ;
end proc:
P := proc(n,k,b)
    local n3,k3 ;
    n3 := convert(n,base,b) ;
    k3 := convert(k,base,b) ;
    Nscatsub(k3,n3) ;
end proc:
A282714 := proc(n,k)
    P(n,k,2) ;
end proc: # R. J. Mathar, Mar 03 2017

Mathematica

nmax = 12;
row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 2], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Dec 14 2017 *)

Extensions

More terms from Lars Blomberg, Mar 03 2017