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A247291
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Number of weighted lattice paths B(n) having no uhd strings.
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4
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1, 1, 2, 4, 7, 15, 32, 69, 154, 346, 786, 1806, 4180, 9745, 22865, 53938, 127865, 304447, 727733, 1745736, 4201350, 10140975, 24544000, 59551327, 144822097, 352940719, 861839226, 2108381480, 5166749329, 12681855551, 31174671514, 76742344774
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OFFSET
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0,3
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COMMENTS
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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: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
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LINKS
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FORMULA
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G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z).
D-finite with recurrence +(n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +3*(n-3)*a(n-4) +(-2*n+9)*a(n-5) +2*(-n+6)*a(n-6) +(n-9)*a(n-8)=0. - R. J. Mathar, Sep 29 2021
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EXAMPLE
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a(4)=7 because we have hhhh, hhH, hHh, Hhh, HH, hud, and udh.
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MAPLE
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eq := G = 1+z*G+z^2*G+z^3*(G-z)*G: G := RootOf(eq, G): Gser := series(G, z = 0, 37): seq(coeff(Gser, z, n), n = 0 .. 35);
# second Maple program:
b:= proc(n, y, t) option remember; `if`(y<0 or y>n or t=3, 0,
`if`(n=0, 1, b(n-1, y, `if`(t=1, 2, 0))+`if`(n>1, b(n-2,
y, 0)+b(n-2, y+1, 1), 0)+b(n-1, y-1, `if`(t=2, 3, 0))))
end:
a:= n-> 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 || t == 3, 0, If[n == 0, 1, b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, 0] + b[n-2, y+1, 1], 0] + b[n-1, y-1, If[t == 2, 3, 0]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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