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A246186
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Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k ascents. 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. An ascent is a maximal sequence of consecutive (1,1)-steps.
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1
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1, 1, 2, 3, 1, 5, 3, 8, 9, 13, 23, 1, 21, 55, 6, 34, 125, 26, 55, 274, 93, 1, 89, 584, 295, 10, 144, 1218, 861, 60, 233, 2496, 2362, 281, 1, 377, 5042, 6176, 1125, 15, 610, 10064, 15542, 4036, 120, 987, 19887, 37906, 13341, 710, 1, 1597, 38963, 90071, 41371, 3479, 21, 2584, 75779, 209357, 121873, 14938, 217, 4181, 146451, 477520, 344160, 58108, 1583, 1
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OFFSET
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0,3
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COMMENTS
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Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
Number of entries in row n is 1 + floor(n/3).
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LINKS
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FORMULA
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The g.f. g = g(t,z) satisfies g = 1 + z*g + z^2*g + t*z^3*g*A, where A = 1 + z*g + z^2*g + z^3*g*A.
The above two equations follow from G(t,x,z) = 1 + z*G(t,1,z) + z^2*G(t,1,z) + t*x*z^3*G(t,1/t,z)*G(t,1,z); here G(t,x,z) is the trivariate g.f. of the paths B(n), where t marks ascents, x marks the fact that the path starts with an (1,1)-step, and z marks weight.
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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 paths in B(3) are hhh, hH, Hh, and UD, having ascents 0, 0, 0, and 1, respectively.
Triangle starts:
1;
1;
2;
3,1;
5,3;
8,9;
13,23,1;
21,55,6;
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MAPLE
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eq := z^3*(-z^2+z^2*t-z+z*t+1)*g^2+(t*z^3-1-z^3+z^2+z)*g+1: g := RootOf(eq, g): gser := simplify(series(g, z = 0, 25)): for n from 0 to 20 do P[n] := sort(coeff(gser, z, n)) end do: for n from 0 to 20 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, t) option remember; `if`(y<0 or y>n, 0,
`if`(n=0, 1, expand(b(n-1, y, 0)+`if`(n>1, b(n-2, y, 0)+
b(n-2, y+1, 1)*`if`(t=0, x, 1), 0) +b(n-1, y-1, 0))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
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MATHEMATICA
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b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y, 0] + If[n>1, b[n-2, y, 0] + b[n-2, y+1, 1]*If[t==0, x, 1], 0] + b[n-1, y-1, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 19 2016, 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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