A138155 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Row sums yield A048285.

T(n,2) = A000071(n+1) (Fibonacci numbers - 1).

T(n,3) = A095681(n-3).

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.

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

EXAMPLE

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, 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;

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 y<k or y<1, 0, b(x-1, y-1, max(y, k),

false))+ `if`(y+2>x, 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<k || y<1, 0, b[x-1, y-1, Max[y, k], False]] + If[y+2 > 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 *)

CROSSREFS