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%I #20 Dec 11 2020 19:15:09
%S 1,1,1,1,4,1,1,7,12,1,1,10,33,36,1,1,13,63,143,108,1,1,16,102,341,609,
%T 324,1,1,19,150,656,1748,2583,972,1,1,22,207,1115,3860,8773,10945,
%U 2916,1,1,25,273,1745,7376,21756,43653,46367,8748,1,1,28,348,2573,12809,45801,119948,216434,196417,26244,1
%N Array read by antidiagonals: T(n,k) = number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1), (-1,1) and whose points lie entirely in the integer rectangle of lattice points {(i, j): 0 <= i <= n, 0 <= j <= k}.
%H Alois P. Heinz, <a href="/A232968/b232968.txt">Antidiagonals n = 0..140, flattened</a>
%H M. Dziemianczuk, <a href="http://dx.doi.org/10.1007/s00373-013-1357-1">Counting Lattice Paths With Four Types of Steps</a>, Graphs and Combinatorics, September 2013, Volume 30, Issue 6, pp 1427-1452.
%F Dziemianczuk gives a g.f.
%e Array begins:
%e 1,1,1,1,1,1,1,1,1,...
%e 1,4,12,36,108,324,972,2916,8748,...
%e 1,7,33,143,609,2583,10945,46367,196417,...
%e 1,10,63,341,1748,8773,43653,216434,1071483,...
%e 1,13,102,656,3860,21756,119948,653612,3539052,...
%e 1,16,150,1115,7376,45801,274243,1606727,9288000,...
%e 1,19,207,1745,12809,86739,558967,3489601,21333553,...
%e ...
%p b:= proc(x, y, m) option remember; `if`(x=0 and y=0, 1,
%p `if`(x>0, b(x-1, y, m), 0)+`if`(y>0, b(x, y-1, m), 0)+
%p `if`(x>0 and y>0, b(x-1, y-1, m), 0)+
%p `if`(x<m and y>0, b(x+1, y-1, m), 0))
%p end:
%p T:= (n, k)-> b(n, k, n):
%p seq(seq(T(d-k, k), k=0..d), d=0..12); # _Alois P. Heinz_, Apr 03 2014
%t b[x_, y_, m_] := b[x, y, m] = If[x == 0 && y == 0, 1, If[x>0, b[x-1, y, m], 0] + If[y>0, b[x, y-1, m], 0] + If[x>0 && y>0, b[x-1, y-1, m], 0] + If[x<m && y>0, b[x+1, y-1, m], 0]]; T[n_, k_] := b[n, k, n]; Table[Table[T[d-k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 24 2014, after _Alois P. Heinz_ *)
%Y Main diagonal gives A339654.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Dec 05 2013
%E More terms from _Alois P. Heinz_, Apr 03 2014
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