A171855 - OEIS

%I #4 Jan 09 2014 20:24:43

%S 1,2,3,5,7,1,10,2,1,15,3,2,1,22,6,3,2,1,32,11,6,3,2,1,47,18,12,6,3,2,

%T 1,69,30,20,13,6,3,2,1,101,50,34,22,14,6,3,2,1,148,81,59,38,24,15,6,3,

%U 2,1,217,130,99,68,42,26,16,6,3,2,1,318,208,163,118,77,46,28,17,6,3,2,1,466

%N Triangle read by rows: T(n,k) is the number of binary words of length n that have no pair of adjacent 1's and have k subwords 0000 (n>=0; k=0 for n=0,1,2; 0<=k<=n-3 for n>=3).

%C Row n has n-2 entries.

%C Sum of entries in row n is the Fibonacci number A000045(n+2).

%C T(n,0)=A003410(n). Sum(k*T(n,k),k>=0)=A004798(n-3) for n>=4.

%F G.f.: G(t,z) = (1+z)(1+z-tz+z^2-tz^2+z^3-tz^3)/(1-tz-z^2+tz^3-z^3+tz^4-z^4).

%e T(5,1)=2 because we have 00001 and 10000; T(7,2)=3 because we have 0000010, 0100000, and 1000001.

%e Triangle starts:

%e 1;

%e 2;

%e 3;

%e 5;

%e 7,1;

%e 10,2,1;

%e 15,3,2,1;

%e 22,6,3,2,1;

%p G := (1+z)*(1+z-t*z+z^2-t*z^2+z^3-t*z^3)/(1-t*z-z^2+t*z^3-z^3+t*z^4-z^4): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 2; 3; for n from 3 to 15 do seq(coeff(P[n], t, k), k = 0 .. n-3) end do; # yields sequence in triangular form

%Y Cf. A000045, A003410, A004798.

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, Feb 15 2010