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

0,8

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

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.

Example

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, ...

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 *)

Crossrefs

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

Sequence in context: A242153 A065066 A266291 * A258829 A110314 A370890

Adjacent sequences: A064042 A064043 A064044 * A064046 A064047 A064048

Keyword

nonn,tabl

Author

Henry Bottomley, Aug 23 2001

Status

approved