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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).

%I A237520 #20 May 10 2014 14:06:18
%S A237520 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,
%T A237520 80,72,80,140,252,140,120,120,140,252,504,280,240,240,280,504,924,504,
%U A237520 420,400,420,504,924,1848,1008,840,800,840,1008,1848,3432,1848,1512,1400,1400,1512,1848
%N 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).
%C A237520 Column k=0 is A063886.
%C A237520 Row sums give A000079.
%H A237520 Alois P. Heinz, <a href="/A237520/b237520.txt">Rows n = 0..200, flattened</a>
%F A237520 G.f. for column k: binomial(2k,k) x^k*A(x) where A(x) is the o.g.f. for A063886.
%e A237520 1;
%e A237520 2;
%e A237520 2,   2;
%e A237520 4,   4;
%e A237520 6,   4,   6;
%e A237520 12,  8,   12;
%e A237520 20,  12,  12,  20;
%e A237520 40,  24,  24,  40;
%e A237520 70,  40,  36,  40,  70;
%e A237520 140, 80,  72,  80,  140;
%e A237520 252, 140, 120, 120, 140, 252;
%e A237520 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.
%p A237520 T:= (n, k)-> 2^irem(n, 2)*binomial(2*k, k)*
%p A237520     binomial(2*iquo(n, 2)-2*k, iquo(n, 2)-k):
%p A237520 seq(seq(T(n, k), k=0..iquo(n, 2)), n=0..14);  # _Alois P. Heinz_, May 10 2014
%t A237520 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
%t A237520 (* or *)
%t A237520 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
%Y A237520 Cf. A067804.
%K A237520 nonn,tabf,walk
%O A237520 0,2
%A A237520 _Geoffrey Critzer_, Feb 08 2014