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A114489
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n that have k valleys at level 1.
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1
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1, 1, 2, 4, 1, 9, 4, 1, 22, 14, 5, 1, 58, 46, 21, 6, 1, 163, 149, 80, 29, 7, 1, 483, 484, 292, 124, 38, 8, 1, 1494, 1589, 1044, 498, 179, 48, 9, 1, 4783, 5288, 3701, 1928, 780, 246, 59, 10, 1, 15740, 17848, 13096, 7304, 3237, 1152, 326, 71, 11, 1, 52956, 61060, 46428
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OFFSET
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0,3
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COMMENTS
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T(n,k) is also the number of Dyck paths of semilength n having k pairs of consecutive valleys at the same level. Example: T(4,1)=4 because we have U(DU)(DU)UDD, U(DU)UD(DU)D, UUD(DU)(DU)D, and UU(DU)(DU)DD, where U=(1,1), D=(1,-1); the pairs of consecutive same-level valleys are shown between parentheses. - Emeric Deutsch, Jun 19 2011
Rows 0 and 1 contain one term each; row n contains n-1 terms (n>=2).
Row sums are the Catalan numbers (A000108).
Sum(k*T(n,k), k=0..n-1) = 6*binomial(2*n-1,n-3)/(n+3) (A003517).
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LINKS
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FORMULA
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G.f.: (1-t*z*C)/((1-z)*(1-t*z*C)-z^2*C), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
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EXAMPLE
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T(4,1) = 4 because we have UU(DU)DDUD, UDUU(DU)DD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the valleys at level 1 are shown between parentheses.
Triangle starts:
1;
1;
2;
4, 1;
9, 4, 1;
22, 14, 5, 1;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=(1-t*z*C)/(1-t*z*C-z+t*z^2*C-z^2*C): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: 1; 1; for n from 2 to 12 do seq(coeff(t*P[n], t^j), j=1..n-1) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, expand(b(x-1, y-1, 1)+
`if`(t=1 and y=1, z, 1)*b(x-1, y+1, 0))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, 1] + If[t == 1 && y == 1, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 20 2015, 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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