

A247297


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


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(n5) (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), 291306.


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*(Gz^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



