

A246181


Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k (1,0)steps of weight 1. 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: a (1,0)step of weight 1; a (1,0)step of weight 2; a (1,1)step of weight 2; a (1,1)step of weight 1. The weight of a path is the sum of the weights of its steps.


1



1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 3, 3, 6, 4, 0, 1, 3, 12, 6, 10, 5, 0, 1, 6, 14, 30, 10, 15, 6, 0, 1, 11, 30, 40, 60, 15, 21, 7, 0, 1, 15, 65, 90, 90, 105, 21, 28, 8, 0, 1, 31, 95, 225, 210, 175, 168, 28, 36, 9, 0, 1, 50, 216, 350, 595, 420, 308, 252, 36, 45, 10, 0, 1
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OFFSET

0,8


COMMENTS

Number of entries in row n is n+1.
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A025250(n+3).
Sum(k*T(n,k), k>=0) = A110320(n) (n>=1).


LINKS

Alois P. Heinz, Rows n = 0..140, flattened
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 + t*z*G + z^2*G + z^3*G^2.


EXAMPLE

Row 3 is 1,2,0,1. Indeed, denoting by h (H) the (1,0)step of weight 1 (2), and u=(1,1), d=(1,1), the four paths of weight 3 are: ud, hH, Hh, and hhh, having 0, 1, 1, and 3 (1,0)steps of weight 1, respectively.
Triangle starts:
1;
0,1;
1,0,1;
1,2,0,1;
1,3,3,0,1;
3,3,6,4,0,1;


MAPLE

eq := G = 1+t*z*G+z^2*G+z^3*G^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0 or y>n, 0,
`if`(n=0, 1, expand(b(n1, y)*x+ `if`(n>1,
b(n2, y)+b(n2, y+1), 0) +b(n1, y1))))
end:
T:= n> (p> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..12); # Alois P. Heinz, Aug 24 2014


MATHEMATICA

b[n_, y_] := b[n, y] = If[y<0  y>n, 0, If[n==0, 1, Expand[b[n1, y]*x + If[n>1, b[n2, y] + b[n2, y+1], 0] + b[n1, y1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* JeanFrançois Alcover, Jun 29 2015, after Alois P. Heinz *)


CROSSREFS

Cf. A004148, A025250, A246177.
Sequence in context: A097559 A293108 A172237 * A123226 A299919 A238270
Adjacent sequences: A246178 A246179 A246180 * A246182 A246183 A246184


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Aug 23 2014


STATUS

approved



