A138155 Triangle read by rows: T(n,k) is the number of Dyck paths with nondecreasing peaks having semilength n and with height of last peak equal to k (1 <= k <= n).
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 19, 13, 5, 1, 1, 20, 42, 37, 19, 6, 1, 1, 33, 89, 97, 62, 26, 7, 1, 1, 54, 183, 240, 184, 95, 34, 8, 1, 1, 88, 368, 570, 511, 312, 137, 43, 9, 1, 1, 143, 728, 1312, 1351, 951, 491, 189, 53, 10, 1
Offset: 1
Examples
T(2,1)=1 because we have /\/\. T(5,4)=4 because we have UDUUUUDDDD, UUDUUUDDDD, UUUDUUDDDD and UUUUDUDDDD, where U=(1,1) and D=(1,-1). Triangle T(n,k) begins: 1; 1, 1; 1, 2, 1; 1, 4, 3, 1; 1, 7, 8, 4, 1; 1, 12, 19, 13, 5, 1; 1, 20, 42, 37, 19, 6, 1; 1, 33, 89, 97, 62, 26, 7, 1; 1, 54, 183, 240, 184, 95, 34, 8, 1;
Links
- Alois P. Heinz, Rows n = 1..141, flattened
- J. G. Penaud and O. Roques, Génération de chemins de Dyck à pics croissants, Discrete Mathematics, Vol. 246, no. 1-3 (2002), 255-267.
Programs
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Maple
g:=sum((-1)^n*t*z^(2*n+1)*(1-z)/(product((1-z)*(1-t*z^i)-z,i=1..n+1)), n=0.. 30): gser:=simplify(series(g,z=0,15)): for n to 11 do P[n]:=sort(coeff(gser, z,n)) end do: for n to 11 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form # second Maple program: b:= proc(x, y, k, t) option remember; `if`(x=0, z^k, `if`(t and yx, 0, b(x-1, y+1, k, true))) end: T:= n-> (p-> seq(coeff(p, z, i), i=1..n))(b(2*n, 0$2, true)): seq(T(n), n=1..12); # Alois P. Heinz, Apr 02 2017 -
Mathematica
b[x_, y_, k_, t_] := b[x, y, k, t] = If[x==0, z^k, If[t && y
x, 0, b[x-1, y+1, k, True]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n}]][b[2*n, 0, 0, True]]; Array[T, 12] // Flatten (* Jean-François Alcover, Jun 19 2018, from Alois P. Heinz's 2nd Maple program *)
Formula
G.f.: Sum_{n >= 0} {(-1)^n tz^{2n+1}(1-z)}/ {Product_{i=1...n+1}((1-z)(1-tz^i)-z)}.
Conjectural g.f.: Sum_{n>=1} (t*x*(1 - x))^n/( Product_{i=2..n+1} (1 - 2*x + x^i) ) = t*x + (t + t^2)*x^2 + (t + 2*t^2 + t^3)*x^3 + ... (checked up to x^12). - Peter Bala, Mar 31 2017
Comments