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A246185
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Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) for which the area between the path and the lines y=0 and y=1 is equal to k. 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: an (1,0)-step of weight 1; an (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.
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1
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1, 1, 2, 3, 1, 5, 2, 1, 8, 5, 3, 1, 13, 10, 8, 5, 1, 21, 20, 18, 14, 8, 1, 34, 38, 39, 35, 26, 12, 1, 55, 71, 80, 80, 70, 49, 17, 1, 89, 130, 160, 174, 169, 142, 90, 23, 1, 144, 235, 312, 365, 385, 363, 290, 158, 30, 1, 233, 420, 598, 745, 840, 861, 785, 588, 264, 38, 1
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
...
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MAPLE
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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)):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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