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A118897 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0000 (n,k>=0).

%I A118897 #16 Jul 23 2024 09:01:19
%S A118897 1,2,4,8,15,1,29,2,1,56,5,2,1,108,12,5,2,1,208,28,12,5,2,1,401,62,29,
%T A118897 12,5,2,1,773,136,65,30,12,5,2,1,1490,294,145,68,31,12,5,2,1,2872,628,
%U A118897 319,154,71,32,12,5,2,1,5536,1328,694,344,163,74,33,12,5,2,1,10671,2787
%N A118897 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0000 (n,k>=0).
%C A118897 Row n has n-2 terms (n>=3). Sum of entries in row n is 2^n (A000079). T(n,0) = A000078(n+4) (tetranacci numbers). T(n,1) = A118898(n). Sum(k*T(n,k),n>=0) = (n-3)*2^(n-4) (A001787).
%H A118897 Alois P. Heinz, <a href="/A118897/b118897.txt">Rows n = 0..143, flattened</a>
%F A118897 G.f.: G(t,z) = [1+(1-t)(z+z^2+z^3)]/[1-(1+t)z-(1-t)(z^2+z^3+z^4)].
%e A118897 T(7,2) = 5 because we have 0000010, 0000011, 0100000, 1100000 and 1000001.
%e A118897 Triangle starts:
%e A118897     1;
%e A118897     2;
%e A118897     4;
%e A118897     8;
%e A118897    15,  1;
%e A118897    29,  2, 1;
%e A118897    56,  5, 2, 1;
%e A118897   108, 12, 5, 2, 1;
%e A118897   ...
%p A118897 G:=(1+(1-t)*(z+z^2+z^3))/(1-(1+t)*z-(1-t)*(z^2+z^3+z^4)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: 1;2;4;8; for n from 4 to 14 do seq(coeff(P[n],t,j),j=0..n-3) od; # yields sequence in triangular form
%p A118897 # second Maple program:
%p A118897 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A118897       expand(b(n-1, min(3, t+1))*`if`(t>2, x, 1))+b(n-1, 0))
%p A118897     end:
%p A118897 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A118897 seq(T(n), n=0..14);  # _Alois P. Heinz_, Sep 17 2019
%t A118897 nn=15;a=x^3/(1-y x)+x+x^2;b=1/(1-x);f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[b (1+a)/(1-a x/(1-x)) ,{x,0,nn}],{x,y}]]//Grid  (* _Geoffrey Critzer_, Nov 18 2012 *)
%Y A118897 Cf. A000079, A000078, A118898, A001787, A076791, A118390.
%K A118897 nonn,tabf
%O A118897 0,2
%A A118897 _Emeric Deutsch_, May 04 2006