cp's OEIS Frontend

A118424 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).

%I A118424 #9 Dec 12 2013 19:14:48
%S A118424 1,2,4,7,1,12,4,20,12,33,30,1,54,68,6,88,144,24,143,291,77,1,232,568,
%T A118424 216,8,376,1080,552,40,609,2012,1318,156,1,986,3688,2988,520,10,1596,
%U A118424 6672,6504,1552,60,2583,11941,13702,4266,275,1,4180,21180,28104,11000
%N A118424 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).
%C A118424 Row n has 1+floor(n/3) terms. Sum of entries in row n is 2^n (A000079). T(n,0)=A000071(n+3)=fibonacci(n+3)-1. T(n,1)=A118425(n). Sum(k*T(n,k),k=0..n-1)=(n-2)*2^(n-3) (A001787).
%H A118424 Alois P. Heinz, <a href="/A118424/b118424.txt">Rows n = 0..250, flattened</a>
%F A118424 G.f.: G(t,z) = 1/[1-2z+(1-t)z^3]. Recurrence relation: T(n,k) = 2T(n-1,k) -T(n-3,k) +T(n-3,k-1) for n>=3.
%e A118424 T(7,2) = 6 because we have 0bb, 1bb, b0b, b1b, bb0 and bb1, where b=001.
%e A118424 Triangle starts:
%e A118424 1;
%e A118424 2;
%e A118424 4;
%e A118424 7,   1;
%e A118424 12,  4;
%e A118424 20, 12;
%e A118424 33, 30, 1;
%p A118424 G:=1/(1-2*z+(1-t)*z^3): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(P[n],t,j),j=0..floor(n/3)) od; # yields sequence in triangular form
%t A118424 nn=15;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2z-(u-1)z^3),{z,0,nn}],{z,u}]]//Grid (* _Geoffrey Critzer_, Dec 03 2013 *)
%Y A118424 Cf. A000071, A000079, A001787, A118425.
%K A118424 nonn,tabf
%O A118424 0,2
%A A118424 _Emeric Deutsch_, Apr 27 2006