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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.

%I A064045 #25 Feb 26 2015 08:19:29
%S A064045 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,
%T A064045 132,5544,4242,740,70,6,1,0,429,56628,73206,16016,1525,102,7,1,0,1430,
%U A064045 613470,1403028,410928,43470,2730,140,8,1,0,4862,6952660,29082339,11925672,1491210,96684,4445,184,9,1
%N 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.
%H A064045 Alois P. Heinz, <a href="/A064045/b064045.txt">Antidiagonals n = 0..140, flattened</a>
%H A064045 R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
%F A064045 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.
%e A064045 Rows start:
%e A064045 1, 0,  0,   0,    0,     0,       0, ...
%e A064045 1, 1,  2,   5,   14,    42,     132, ...
%e A064045 1, 2, 10,  70,  588,  5544,   56628, ...
%e A064045 1, 3, 24, 285, 4242, 73206, 1403028, ...
%p A064045 a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A064045        add(binomial(2*k, 2*j)*binomial(2*j, j)/
%p A064045        (j+1)*a(n-1, k-j), j=0..k))
%p A064045     end:
%p A064045 seq(seq(a(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, May 06 2014
%t A064045 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_ *)
%Y A064045 Rows include A000007, A000108, A005568, A064037. Columns include A000012, A001477, A049450, A064046. Cf. A064044.
%K A064045 nonn,tabl
%O A064045 0,8
%A A064045 _Henry Bottomley_, Aug 23 2001