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A246179
Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k returns to the horizontal axis (i.e., (1,-1)-steps ending on the horizontal axis). The members of B(n) are paths of weight n that start in (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
1
1, 1, 2, 3, 1, 5, 3, 8, 9, 13, 23, 1, 21, 56, 5, 34, 131, 20, 55, 300, 67, 1, 89, 678, 204, 7, 144, 1523, 581, 35, 233, 3416, 1580, 143, 1, 377, 7677, 4155, 517, 9, 610, 17329, 10663, 1716, 54, 987, 39353, 26880, 5352, 259, 1, 1597, 90000, 66891, 15924, 1079
OFFSET
0,3
COMMENTS
Number of entries in row n is 1+floor(n/3).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0)=A000045(n+1) (the Fibonacci numbers).
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 + t*z^3*g*G, where g=1+z*g+z^2*g+z^3*g^2.
EXAMPLE
Row 3 is 3,1. 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 1, 0, 0, and 0 returns to the horizontal axis, respectively.
Triangle starts:
1;
1;
2;
3,1;
5,3;
8,9;
13,23,1;
MAPLE
eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): G := 1/(1-z-z^2-t*z^3*g): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*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(n-1, y)+`if`(n>1, b(n-2, y)+
b(n-2, y+1), 0) +b(n-1, y-1)*`if`(y=1, x, 1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20); # 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[n-1, y] + If[n>1, b[n-2, y] + b[n-2, y+1], 0] + b[n-1, y-1]*If[y==1, x, 1]]]]; T[n_] := Function[ {p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 23 2014
STATUS
approved