A247297 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uudd strings.

1, 1, 2, 4, 8, 17, 36, 1, 80, 2, 180, 5, 410, 13, 946, 32, 2203, 80, 5173, 199, 1, 12233, 499, 3, 29108, 1255, 9, 69643, 3161, 28, 167437, 7984, 81, 404311, 20206, 231, 980125, 51228, 650, 1, 2384441, 130090, 1812, 4, 5819576, 330835, 5016, 14

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

Table of n, a(n) for n=0..47.

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

Cf. A004148, A110320, A247298.

Sequence in context: A098083 A182900 A202843 * A292322 A008999 A052903

Adjacent sequences: A247294 A247295 A247296 * A247298 A247299 A247300

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 17 2014

Status

approved