%I #46 Aug 26 2018 19:30:32
%S 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,
%T 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,
%U 1,3,0,2,0,1,0,0,0,1,1,2,4,1,2,0,2,0,0
%N Base-2 generalized Pascal triangle P_2 read by rows (see Comments for precise definition).
%C 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.
%C Row n has sum n+1.
%H Lars Blomberg, <a href="/A282714/b282714.txt">Table of n, a(n) for n = 0..10000</a>
%H Julien Leroy, Michel Rigo, Manon Stipulanti, <a href="http://dx.doi.org/10.1016/j.disc.2017.01.003">Counting the number of non-zero coefficients in rows of generalized Pascal triangles</a>, Discrete Mathematics 340 (2017), 862-881.
%H Julien Leroy, Michel Rigo, Manon Stipulanti, <a href="https://arxiv.org/abs/1705.10065">Counting Subwords Occurrences in Base-b Expansions</a>, arXiv:1705.10065 [math.CO], 2017.
%H Julien Leroy, Michel Rigo, Manon Stipulanti, <a href="http://math.colgate.edu/~integers/sjs13/sjs13.Abstract.html">Counting Subwords Occurrences in Base-b Expansions</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
%H Manon Stipulanti, <a href="https://arxiv.org/abs/1801.03287">Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems</a>, arXiv:1801.03287 [math.CO], 2018.
%e Triangle begins:
%e 1,
%e 1,1,
%e 1,1,1,
%e 1,2,0,1,
%e 1,1,2,0,1,
%e 1,2,1,1,0,1,
%e 1,2,2,1,0,0,1,
%e 1,3,0,3,0,0,0,1,
%e 1,1,3,0,3,0,0,0,1
%e 1,2,2,1,1,2,0,0,0,1
%e 1,2,3,1,1,1,1,0,0,0,1
%e 1,3,1,3,0,2,0,1,0,0,0,1
%e 1,2,4,1,2,0,2,0,0,0,0,0,1
%e ...
%e The binary numbers are epsilon, 1, 10, 11, 100, 101, 110, 111, 1000, ...
%e The fifth number 101 contains
%e eps 1 10 11 100 101 respectively
%e .1..2..1..1...0...1 times, which is row 5 of the triangle.
%p Nscatsub := proc(subw,w)
%p local lsubw,lw,N,wri,wr,i ;
%p lsubw := nops(subw) ;
%p lw := nops(w) ;
%p N := 0 ;
%p if lsubw = 0 then
%p return 1 ;
%p elif lsubw > lw then
%p return 0 ;
%p else
%p for wri in combinat[choose](lw,lsubw) do
%p wr := [] ;
%p for i in wri do
%p wr := [op(wr),op(i,w)] ;
%p end do:
%p if verify(subw,wr,'sublist') then
%p N := N+1 ;
%p end if;
%p end do:
%p end if;
%p return N ;
%p end proc:
%p P := proc(n,k,b)
%p local n3,k3 ;
%p n3 := convert(n,base,b) ;
%p k3 := convert(k,base,b) ;
%p Nscatsub(k3,n3) ;
%p end proc:
%p A282714 := proc(n,k)
%p P(n,k,2) ;
%p end proc: # _R. J. Mathar_, Mar 03 2017
%t nmax = 12;
%t row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 2], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];
%t Table[row[n], {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Dec 14 2017 *)
%Y A007306 gives (essentially) the number of nonzero entries in the rows.
%K nonn,tabl
%O 0,8
%A _N. J. A. Sloane_, Mar 02 2017
%E More terms from _Lars Blomberg_, Mar 03 2017