%I #38 Aug 12 2020 11:16:33
%S 1,1,1,1,2,1,1,2,2,1,1,2,4,2,1,1,2,5,5,2,1,1,2,6,8,6,2,1,1,2,7,11,11,
%T 7,2,1,1,2,8,14,18,14,8,2,1,1,2,9,17,26,26,17,9,2,1,1,2,10,20,35,42,
%U 35,20,10,2,1,1,2,11,23,45,62,62,45,23,11,2,1,1,2,12,26,56,86,100,86,56,26,12,2,1
%N Array T(m, n) read by antidiagonals: number of binary strings with m 1's and n 0's without zigzags.
%H E. Munarini and N. Z. Salvi, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s49zagaglia.html">Binary strings without zigzags</a>, [<a href="http://www.emis.de/journals/SLC/wpapers/s49zagaglia.html">alternative link</a>], Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
%H R. Pemantle and M. C. Wilson, <a href="https://arxiv.org/abs/math/0512548">Twenty combinatorial examples of asymptotics derived from multivariate generating functions</a>, arXiv:math/0512548 [math.CO], 2005-2007.
%H R. Pemantle and M. C. Wilson, <a href="http://dx.doi.org/10.1137/050643866">Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-372. See p. 269
%F G.f.: (1 + x*y + x^2*y^2) / (1 - x - y + x*y - x^2*y^2).
%F T(m, n) = Sum{k=0..min(m+[m/2], n+[n/2]), C(m-k+2[k/3], [k/3])*C(n-k+2[k/3], [k/3]) }.
%e Array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1,
%e 1, 2, 2, 2, 2, 2, 2, 2,
%e 1, 2, 4, 5, 6, 7, 8, 9,
%e 1, 2, 5, 8, 11, 14, 17, 20,
%e 1, 2, 6, 11, 18, 26, 35, 45,
%e 1, 2, 7, 14, 26, 42, 62, 86,
%e 1, 2, 8, 17, 35, 62, 100, 150,
%e 1, 2, 9, 20, 45, 86, 150, 242,
%p gf:=(1 + x*y + x^2*y^2)/(1 - x - y + x*y - x^2*y^2);seq(seq(coeff(series(coeff(series(gf,y,m+1),y,m),x,d-m+1),x,d-m), m=0..d), d=0..9);
%t T[m_, n_] := Sum[Binomial[m - k + 2 Floor[k/3], Floor[k/3]] Binomial[n - k + 2 Floor[k/3], Floor[k/3]], {k, 0, Min[m+Floor[m/2], n+Floor[n/2]]}];
%t Table[T[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Aug 17 2018 *)
%o (PARI) T(m,n)=sum(k=0,min(m+m\2,n+n\2),binomial(m-k+2*(k\3),k\3)*binomial(n-k+2*(k\3),k\3))
%o (PARI) T(n,k) = {x = xx + xx*O(xx^n); y = yy + yy*O(yy^k); polcoeff(polcoeff((1 + x*y + x^2*y^2)/(1 - x - y + x*y - x^2*y^2), n, xx), k, yy);} \\ _Michel Marcus_, Nov 25 2013
%o (PARI) {A(n, m) = if( n<0 || m<0, 0, polcoeff( polcoeff( (1 + x*y + x^2*y^2 ) / (1 - x - y + x*y - x^2*y^2) + x * O(x^n), n) + y * O(y^m), m))}; /* _Michael Somos_, Jun 06 2016 */
%Y Main diagonal is A078678. Antidiagonal sums are A128588.
%K nonn,tabl
%O 0,5
%A _Ralf Stephan_, Oct 10 2004