OFFSET
0,3
COMMENTS
The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
Apparently, number of terms in row n is 1+floor(n^2/8).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
T(n,1) = A001629(n-1) (n>=1).
LINKS
Alois P. Heinz, Rows n = 0..65, flattened
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
FORMULA
The trivariate g.f. G=G(t,s,z), where t marks area, s marks length (=number of steps), and z marks weight, satisfies G = 1+szG+sz^2G+ts^2z^3G(t,ts,z)G. This follows at once from the fact that every nonempty path is of the form hC or HC or UCDC, where h denotes a (1,0)-step of weight 1, H denotes a (1,0)-step of weight 2, U denotes a (1,1)-step, D denotes a (1,-1)-step, and the C's denote paths, not necessarily the same. From the equation one can find G(t,s,z) as a continued fraction (the Maple program makes use of this).
EXAMPLE
Row 3 is 3,1; indeed, B(3) consists of the paths hhh, hH, Hh, UD with areas 0,0,0,1, respectively.
Triangle starts:
1;
1;
2;
3, 1;
5, 2, 1;
8, 5, 3, 1;
13, 10, 8, 4, 2;
21, 20, 18, 12, 7, 3, 1;
34, 38, 39, 30, 22, 12, 7, 2, 1;
55, 71, 80, 70, 57, 39, 26, 14, 7, 3, 1;
89, 130, 160, 154, 138, 106, 81, 52, 34, 18, 10, 4, 2;
MAPLE
g := 1/(1-z-z^2-t*z^3*A[1]): for j to 15 do A[j] := 1/(1-t^j*z-t^j*z^2-t^(2*j+1)*z^3*A[j+1]) end do: gser := simplify(series(g, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/8)*n^2)) 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)*x^y +`if`(n>1, b(n-2, y)*x^y+b(n-2, y+1)*
x^(y+1/2), 0) +b(n-1, y-1)*x^(y-1/2))))
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 20 2014
MATHEMATICA
b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n == 0, 1, Expand[b[n-1, y]*x^y + If[n>1, b[n-2, y]*x^y + b[n-2, y+1]*x^(y+1/2), 0] + b[n-1, y-1]*x^(y-1/2)]]]; 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, May 27 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 20 2014
STATUS
approved