A064045 Square array read by antidiagonals of number of length 2k walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 10, 3, 1, 0, 14, 70, 24, 4, 1, 0, 42, 588, 285, 44, 5, 1, 0, 132, 5544, 4242, 740, 70, 6, 1, 0, 429, 56628, 73206, 16016, 1525, 102, 7, 1, 0, 1430, 613470, 1403028, 410928, 43470, 2730, 140, 8, 1, 0, 4862, 6952660, 29082339, 11925672, 1491210, 96684, 4445, 184, 9, 1

Offset: 0

Author

Henry Bottomley, Aug 23 2001

Keywords

• nonn
• tabl

Examples

			Rows start:
1, 0,  0,   0,    0,     0,       0, ...
1, 1,  2,   5,   14,    42,     132, ...
1, 2, 10,  70,  588,  5544,   56628, ...
1, 3, 24, 285, 4242, 73206, 1403028, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Crossrefs

Rows include A000007, A000108, A005568, A064037. Columns include A000012, A001477, A049450, A064046. Cf. A064044.

Programs

Maple

a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
       add(binomial(2*k, 2*j)*binomial(2*j, j)/
       (j+1)*a(n-1, k-j), j=0..k))
    end:
seq(seq(a(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014

Mathematica

a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[2*k, 2*j]* Binomial[2*j, j]/(j+1)*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

Formula

a(n,k) = Sum_{j=0..k} C(2k,2j) c(j) a(n-1,k-j) where c(j) = C(2j,j)/(j+1) = A000108(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.