A118920 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross the x-axis k times (n>=1, k>=0).

2, 4, 2, 10, 8, 2, 28, 28, 12, 2, 84, 96, 54, 16, 2, 264, 330, 220, 88, 20, 2, 858, 1144, 858, 416, 130, 24, 2, 2860, 4004, 3276, 1820, 700, 180, 28, 2, 9724, 14144, 12376, 7616, 3400, 1088, 238, 32, 2, 33592, 50388, 46512, 31008, 15504, 5814, 1596, 304, 36, 2

Offset: 1

Author

Emeric Deutsch, May 06 2006

Keywords

• nonn
• tabl

Comments

A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1).

Row sums are the central binomial coefficients (A000984). T(n,0)=2*A000108(n) (the Catalan numbers doubled). T(n,1)=2*A002057(n-2). Sum(k*T(n,k),k>=0)=2*A008549(n-1). For crossings of the x-axis in one direction, see A118919.

This triangle is related to paired pattern P_3 and P_4 defined in the Pan & Remmel link. - Ran Pan, Feb 01 2016

Examples

			T(3,1)=8 because we have ud|dudu,ud|dduu,udud|du,uudd|du,du|udud,du|uudd, dudu|ud and dduu|ud (the crossings of the x-axis are shown by |).
Triangle starts:
   2;
   4, 2;
  10, 8, 2;
  28,28,12, 2;
  84,96,54,16,2;
		

Links

• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A000984, A000108, A002057, A008549, A039598, A118919.

Programs

Maple

T:=(n,k)->2*(k+1)*binomial(2*n,n-k-1)/n: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

Mathematica

Table[2 (k + 1) Binomial[2 n, n - k - 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Feb 01 2016 *)

PARI

T(n,k) = 2*(k+1)*binomial(2*n,n-k-1)/n \\ Charles R Greathouse IV, Feb 01 2016

Sage

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A118920_triangle(n) :
    D = [0]*(n+2); D[1] = 2
    b = True; h = 1
    for i in range(2*n) :
        if b :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
        if b : print([D[z] for z in (1..h-1)])
A118920_triangle(10)  # Peter Luschny, Oct 19 2012

Formula

T(n,k) = 2*(k+1)*binomial(2*n,n-k-1)/n.

G.f.: G(t,z)=2*z*C^2/(1-t*z*C^2), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.

More generally, the trivariate g.f. G=G(x,y,z), where x (y) marks number of downward (upward) crossings of the x-axis, is given by G = z*C^2*(2+(x+y)*z*C^2)/(1-x*y*z^2*C^4).

a(n) = 2 * A039598(n-1). - Georg Fischer, Oct 27 2021