%I #92 Oct 21 2023 11:28:53

%S 1,1,1,2,2,2,4,5,5,4,8,12,14,12,8,16,28,37,37,28,16,32,64,94,106,94,

%T 64,32,64,144,232,289,289,232,144,64,128,320,560,760,838,760,560,320,

%U 128,256,704,1328,1944,2329,2329,1944,1328,704,256,512,1536,3104,4864,6266

%N Square array read by antidiagonals: T(m,n) = Sum_{k=1..m-1} T(m-k,n) + Sum_{k=1..n-1} T(m,n-k).

%C T(m,n) is the sum of all the entries above it plus the sum of all the entries to the left of it.

%C T(m,n) equals the number of ways to move a chess rook from the lower left corner to square (m,n), with the rook moving only up or right. - _Francisco Santos_, Oct 20 2005

%C T(m,n) is the number of nim games that start with two piles of stones of sizes m and n. - Martin J. Erickson (erickson(AT)truman.edu), Dec 05 2008

%C The same sequences arises from reading the following triangle by rows: Start with 1, then use a Pascal-like rule, where each new entry is the sum of all terms in the two diagonals that converge at that point. See example below. - _J. M. Bergot_, Jun 08 2013

%C T(n,k) is odd iff (n,k) = (1,1), k = n-1, or k = n+1. - _Peter Kagey_, Apr 20 2020

%H Peter Kagey, <a href="/A035002/b035002.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals, flattened)

%H C. Coker, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00037-2">Enumerating a class of lattice paths</a>, Discrete Math., 271 (2003), 13-28.

%H M. Erickson, S. Fernando, and K. Tran, <a href="https://www.semanticscholar.org/paper/Enumerating-Rook-and-Queen-Paths-Erickson-Fernando/fc8d32756ec73ccae8b28ad93431c13c571c6f10">Enumerating Rook and Queen Paths</a>, Bulletin of the Institute for Combinatorics and Its Applications, Volume 60 (2010), 37-48.

%F G.f. T(n; x) for n-th row satisfies: T(n; x) = Sum_{k=1..n} (1+x^k)*T(n-k; x), T(0; x) = 1. - _Vladeta Jovovic_, Sep 03 2002

%F T(m+1,n+1) = 2*T(m+1,n) + 2*T(m,n+1) - 3*T(m,n); T(n,1) = T(1,n) = A011782(n). - _Francisco Santos_, Oct 20 2005

%F G.f.: ((x-1)*y-x+1)/((3*x-2)*y-2*x+1). - _Vladimir Kruchinin_, Apr 14 2015

%F T(n,m) = Sum_{i=0..m} C(m-1,m-i)*Sum_{k=0..n} C(k+i,i)*C(n-1,n-k). - _Vladimir Kruchinin_, Apr 14 2015

%F T(n,m) = T(m,n) for all n and m. - _Michael Somos_, Oct 04 2023

%e Table begins:

%e 1 1 2 4 8 16 32 64 ...

%e 1 2 5 12 28 64 144 320 ...

%e 2 5 14 37 94 232 560 1328 ...

%e 4 12 37 106 289 760 1944 4864 ...

%e Alternative construction as a triangle:

%e 1

%e 1 1

%e 2 2 2

%e 4 5 5 4

%e 8 12 14 12 8

%e 16 28 37 37 28 16

%p A035002 := proc(m,n)

%p option remember;

%p if n = 1 and m= 1 then

%p 1;

%p elif m = 1 then

%p 2^(n-2) ;

%p elif n = 1 then

%p 2^(m-2) ;

%p else

%p add( procname(m-k,n),k=1..m-1) + add( procname(m,n-k),k=1..n-1) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 06 2013

%t T[n_, 1] = 2^(n-2); T[1, n_] = 2^(n-2); T[1, 1] = 1; T[m_, n_] := T[m, n] = Sum[T[m-k, n], {k, 1, m-1}] + Sum[T[m, n-k], {k, 1, n-1}]; Flatten[Table[T[m-n+1 , n], {m, 1, 11}, {n, 1, m}]] (* _Jean-François Alcover_, Nov 04 2011 *)

%t nMax = 11; T = (((x - 1)*y - x + 1)/((3*x - 2)*y - 2*x + 1) + O[x]^nMax // Normal // Expand) + O[y]^nMax // Normal // Expand // CoefficientList[#, {x, y}]&; Table[T[[n - k + 1, k]], {n, 1, nMax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 18 2018, after _Vladimir Kruchinin_ *)

%t T[ n_, m_] := SeriesCoefficient[ (1 - x)*(1 - y)/( 1 - 2*x - 2*y + 3*x*y), {x, 0, n}, {y, 0, m}]; (* _Michael Somos_, Oct 05 2023 *)

%o (Maxima)

%o T(n,m):=sum(binomial(m-1,m-i)*sum(binomial(k+i,i)*binomial(n-1,n-k),k,0,n),i,0,m); /* _Vladimir Kruchinin_, Apr 14 2015 */

%Y Cf. A035001, A051708, A025192 (antidiagonal sums).

%K nonn,tabl,easy,nice

%O 1,4

%A _Erich Friedman_