A237520 Irregular triangular array read by rows: T(n,k) is the number of n-step walks (steps +1,-1) on the x-axis beginning at the origin that are on the origin for the last time on step 2k, n>=0, 0<=k<=floor(n/2).

1, 2, 2, 2, 4, 4, 6, 4, 6, 12, 8, 12, 20, 12, 12, 20, 40, 24, 24, 40, 70, 40, 36, 40, 70, 140, 80, 72, 80, 140, 252, 140, 120, 120, 140, 252, 504, 280, 240, 240, 280, 504, 924, 504, 420, 400, 420, 504, 924, 1848, 1008, 840, 800, 840, 1008, 1848, 3432, 1848, 1512, 1400, 1400, 1512, 1848

Offset: 0

Author

Geoffrey Critzer, Feb 08 2014

Keywords

• nonn
• tabf
• walk

Comments

Column k=0 is A063886.

Row sums give A000079.

Examples

			1;
2;
2,   2;
4,   4;
6,   4,   6;
12,  8,   12;
20,  12,  12,  20;
40,  24,  24,  40;
70,  40,  36,  40,  70;
140, 80,  72,  80,  140;
252, 140, 120, 120, 140, 252;
T(4,1) = 4 because we have: (-1,+1,-1,-1), (-1,+1,+1,+1), (+1,-1,-1,-1), (+1,-1,+1,+1). These walks have 4 steps and are on the origin for the last time on step 2*1=2.
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A067804.

Programs

Maple

T:= (n, k)-> 2^irem(n, 2)*binomial(2*k, k)*
    binomial(2*iquo(n, 2)-2*k, iquo(n, 2)-k):
seq(seq(T(n, k), k=0..iquo(n, 2)), n=0..14);  # Alois P. Heinz, May 10 2014

Mathematica

nn=20;d=(1-(1-4x^2)^(1/2))/(2x^2);Map[Select[#,#>0&]&,Transpose[Table[ CoefficientList[Series[Binomial[2n,n]x^(2n)(1-2x^2d)/(1-2x),{x,0,nn}],x],{n,0,nn/2}]]]//Grid
(* or *)
f[list_]:=If[Max[Flatten[Position[list,0]]]== -Infinity,0,Max[Flatten[ Position[list,0]]]]; Table[Distribution[Map[f,Map[Accumulate, Strings[{-1,1},n]]]],{n,0,10}]//Grid

Formula

G.f. for column k: binomial(2k,k) x^k*A(x) where A(x) is the o.g.f. for A063886.