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A135333
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k DUDU's starting at level 1.
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2
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1, 1, 2, 4, 1, 11, 2, 1, 32, 7, 2, 1, 99, 22, 8, 2, 1, 318, 73, 26, 9, 2, 1, 1051, 246, 90, 30, 10, 2, 1, 3550, 844, 312, 108, 34, 11, 2, 1, 12200, 2936, 1096, 384, 127, 38, 12, 2, 1, 42520, 10334, 3886, 1379, 462, 147, 42, 13, 2, 1, 149930, 36736, 13897, 4978, 1694
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OFFSET
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0,3
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COMMENTS
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Each of rows 0, 1 and 2 contains one entry. Row n contains n-1 entries (n >= 2). Row sums are the Catalan numbers (A000108). Column 0 yields A135339. - Emeric Deutsch, Dec 13 2007
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LINKS
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FORMULA
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G.f.: 1+zC+z^2*C^3/[1+(1-t)zC], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). T(n,k) = d(0,k)*c(n-1)+Sum[(-1)^(j-k)*(j+3)binomial(j,k)binomial(2n-j-2,n),j=k..n-2]/(n+1), where c(m) = binomial(2m,m)/(m+1) = A000108(m) is a Catalan number and d(0,k) is the Kronecker symbol. - Emeric Deutsch, Dec 13 2007
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EXAMPLE
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Triangle begins:
1
1
2
4 1
11 2 1
32 7 2 1
99 22 8 2 1
318 73 26 9 2 1
1051 246 90 30 10 2 1
...
T(4,1)=2 because we have U(DUDU)UDD and UUD(DUDU)D; T(4,2)=1 because we have U(DU[DU)DU]D (the DUDU's starting at level 1 are shown between parentheses).
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MAPLE
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G:=1+z*C+z^2*C^3/(1+(1-t)*z*C): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=simplify(series(G, z=0, 17)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j=0..n-2) end do; # yields sequence in triangular form; Emeric Deutsch, Dec 13 2007
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,
`if`(y<x, b(x-1, y+1, [1, 3, 1, 3][t])*`if`(t=4, z, 1), 0)+
`if`(y>0, b(x-1, y-1, `if`(y=1, [2, 1, 4, 1][t], 1)), 0)))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = Expand[If[x == 0, 1, If[y < x, b[x - 1, y + 1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1], 0] + If[y > 0, b[x - 1, y - 1, If[y == 1, {2, 1, 4, 1}[[t]], 1]], 0]]]; T[n_] := Function [p, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 14 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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EXTENSIONS
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STATUS
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approved
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