OFFSET
0,3
COMMENTS
B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
Row n contains 1 + floor(n/6) entries.
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A247298(n).
Sum(k*T(n,k), k=0..n) =A110320(n-5) (n>=6)
LINKS
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
FORMULA
G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z^3 + t*z^3).
EXAMPLE
T(6,1)=1 because among the 37 (=A004148(7)) paths in B(6) only uudd contains uudd.
T(13,2)=3 because we have huudduudd, uuddhuudd, and uudduuddh.
Triangle starts:
1;
1;
2;
4;
8;
17;
36,1;
80,2;
MAPLE
eq := G = 1+z*G+z^2*G+z^3*(G-z^3+t*z^3)*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, k), k = 0 .. floor((1/6)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 17 2014
STATUS
approved