OFFSET
0,4
COMMENTS
Row n has 1+ceiling(n/2) terms.
FORMULA
G.f.: G(t,z) = (1+z)(1+tz-tz^2)/(1-(2+t)z^2+tz^4). The trivariate generating function H(t,s,z), where t marks number of 0's in odd position and s marks number of 0's in even position, is given by H(t,s,z) = (1+(1+t)z-tsz^3)/(1-(1+t+s)z^2+tsz^4).
Row sums are the Fibonacci numbers (A000045).
T(2n,k) = T(2n-1,k) + T(2n-2,k) (n >= 1).
T(2n,k) = A129721(2n,k).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A129720(n).
EXAMPLE
T(6,2)=4 because we have 110101, 011101, 010110 and 010111.
Triangle starts:
1;
1, 1;
2, 1;
2, 2, 1;
4, 3, 1;
4, 5, 3, 1;
8, 8, 4, 1;
MAPLE
G:=(1+z)*(1+t*z-t*z^2)/(1-(2+t)*z^2+t*z^4): Gser:=simplify(series(G, z=0, 20)): for n from 0 to 17 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 17 do seq(coeff(P[n], t, j), j=0..ceil(n/2)) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 13 2007
STATUS
approved