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%I #63 Sep 08 2022 08:45:13
%S 5,13,68,25,222,1110,41,555,3951,19010,61,1171,11263,70438,329126,85,
%T 2198,27468,216618,1245986,5693968,113,3788,59676,579330,4022546,
%U 21832492,98074332,145,6117,118605,1389927,11462495,72887139,379145115,1680306750
%N Triangle T read by rows: T(m,n) = number of convex polyominoes with an m+1 X n+1 minimal bounding rectangle, m > 0, n <= m.
%H Michael De Vlieger, <a href="/A093118/b093118.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened).
%H Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, André Schulz, <a href="https://arxiv.org/abs/1903.01095">The Number of Convex Polyominoes with Given Height and Width</a>, arXiv:1903.01095 [math.CO], 2019.
%H Ira Gessel, <a href="http://www.labmath.uqam.ca/~annales/volumes/24-1/PDF/063-066.pdf">On the number of convex polyominoes</a>, Annales des Sciences Mathématiques du Québec, 24 (2000), 63-66.
%H V. J. W. Guo and J. Zeng, <a href="https://arxiv.org/abs/math/0403262">The number of convex polyominoes and the generating function of Jacobi polynomials</a>, arXiv:math/0403262 [math.CO], 2004.
%F T(m,n) = ((m+n+m*n)*C(2*m+2*n, 2*m) - 2*m*n*C(m+n, m)^2)/(m+n), for m + n > 0.
%F T(m,n) = C(2*m+2*n,2*m) + ((2*m+2*n-1)/2)*C(2*m+2*n-2,2*m-1) - 2*(m+n-1) *C(m+n,m)*C(m+n-2,m-1), for m >= 0, n >= 0. - _Günter Rote_, Feb 12 2019
%e Triangle begins:
%e 5,
%e 13, 68,
%e 25, 222, 1110,
%e 41, 555, 3951, 19010,
%e 61, 1171, 11263, 70438, 329126,
%e 85, 2198, 27468, 216618, 1245986, 5693968,
%e ...
%e This is the lower half of an infinite square table that is symmetric at the main diagonal (T(m,n)=T(n,m)).
%e From _Günter Rote_, Feb 12 2019: (Start)
%e For m=2 and n=1, the T(2,1)=13 polyominoes in a 3 X 2 rectangle are the five polyominoes
%e .
%e +---+---+---+ +---+ +---+---+
%e | | | | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+---+
%e | | | | | | | | | | |
%e +---+---+---+ +---+---+---+ +---+---+
%e .
%e +---+ +---+---+
%e | | | | |
%e +---+---+---+ +---+---+---+
%e | | | | | | | |
%e +---+---+---+ +---+---+---+
%e .
%e plus all their different horizontal and vertical reflections (1 + 2 + 2 + 4 + 4 = 13 polyominoes in total). (End)
%p T:= (m, n)-> (m+n+m*n)/(m+n)*binomial(2*m+2*n, 2*m)
%p -2*m*n/(m+n)*binomial(m+n, m)^2:
%p seq(lprint(seq(T(m, n), n=1..m)), m=1..10); # _Alois P. Heinz_, Feb 24 2019
%t T[m_, n_] := (m+n+m n)/(m+n) Binomial[2m + 2n, 2m] - 2 m n/(m+n) Binomial[ m+n, m]^2;
%t Table[T[m, n], {m, 1, 8}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Aug 17 2018 *)
%o (Sage)
%o def T(m,n):
%o w, h = m+1, n+1 # width and height
%o p = w+h # half the perimeter
%o return ( binomial(2*p-4, 2*w-2) + binomial(2*p-6, 2*w-3)*(p-5/2) - 2*(p-3)*binomial(p-2, w-1)*binomial(p-4, w-2) ) # _Günter Rote_, Feb 13 2019
%o (PARI) {T(n,k) = ((n+k+n*k)*binomial(2*n+2*k, 2*n) - 2*n*k*binomial(n+k, n)^2)/(n+k)};
%o for(n=1,8, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 18 2019
%o (Magma) [[((n+k+n*k)*Binomial(2*n+2*k, 2*n) - 2*n*k*Binomial(n+k, n)^2)/(n+k): k in [1..n]]: n in [1..8]]; // _G. C. Greubel_, Feb 18 2019
%Y Columns T(m, 1) = A001844(m), T(m, 2) = A093119(m). Diagonal T(n, n) = A093120(n).
%Y Sums of T(m,n) with fixed sum m+n (including entries with n > m and the trivial ones: T(0,x)=T(y,0)=1), are A005436. - _Günter Rote_, Feb 12 2019
%K nonn,tabl,easy,walk
%O 1,1
%A _Ralf Stephan_, Mar 21 2004
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