OFFSET
0,3
COMMENTS
Number of entries in row n is n-1 (n>=2).
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). - Robert Israel, Aug 28 2014
LINKS
Alois P. Heinz, Rows n = 0..150, 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
G.f. g = g(t,z) satisfies (t*z^3 + z^2 - t*z^2 + z - t*z - 1 + t)*g^2 +(1 - 2*t + t*z + t*z^2)*g + t = 0.
The above equation for g = G(t,1,z) follows from the trivariate g.f. G(t,w,z), where z marks weight, t marks the considered area, and w marks the length of the path (= number of steps) and which satisfies G(t,w,z) = 1 + w*z*G(t,w,z) + w*z^2*G(t,w,z) + t*w^2*z^3*G(t,w,z)*G(1,tw,z).
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; the areas to be considered are 1, 0, 0, and 0, respectively.
Triangle starts:
1;
1;
2;
3, 1;
5, 2, 1;
8, 5, 3, 1;
13,10, 8, 5, 1;
...
MAPLE
eq := (t*z^3+z^2-t*z^2+z-t*z-1+t)*g^2+(t*z^2+t*z+1-2*t)*g+t = 0: g := RootOf(eq, g, 1): gser := simplify(series(g, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, j), j = 0 .. 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^min(1, y)+`if`(n>1, b(n-2, y)*x^min(1, y)+
b(n-2, y+1)*x^min(y+1/2, 1), 0)+b(n-1, y-1)*x^min(y-1/2, 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 28 2014
MATHEMATICA
b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y] * x^Min[1, y] + If[n>1, b[n-2, y]*x^Min[1, y] + b[n-2, y+1]*x^Min[y+1/2, 1], 0] + b[n-1, y-1]*x^Min[y-1/2, 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, Feb 08 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 28 2014
STATUS
approved