%I #23 May 04 2021 09:01:18
%S 1,1,1,1,9,1,1,36,36,1,1,100,400,100,1,1,225,2500,2500,225,1,1,441,
%T 11025,30625,11025,441,1,1,784,38416,240100,240100,38416,784,1,1,1296,
%U 112896,1382976,3111696,1382976,112896,1296,1,1,2025,291600,6350400,28005264,28005264,6350400,291600,2025,1
%N Triangle read by rows: T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
%H Stefano Spezia, <a href="/A174158/b174158.txt">First 150 rows of the triangle, flattened</a>
%H Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, <a href="https://arxiv.org/abs/2105.00971">Enumeration of parallelogram polycubes</a>, arXiv:2105.00971 [cs.DM], 2021.
%F T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
%F T(n,m) = A001263(n,m)^2.
%F T(n,m) = A000290(A007318(n - 1, m - 1)*A007318(n, m - 1)/m). - _Stefano Spezia_, Dec 23 2018
%e n\m | 1 2 3 4 5 6 7
%e ----|--------------------------------------------------------------
%e 1 | 1
%e 2 | 1 1
%e 3 | 1 9 1
%e 4 | 1 36 36 1
%e 5 | 1 100 400 100 1
%e 6 | 1 225 2500 2500 225 1
%e 7 | 1 441 11025 30625 11025 441 1
%p a := (n, m) -> binomial(n-1, m-1)^2*binomial(n, m-1)^2/m^2: seq(seq(a(n, m), m = 1 .. n), n = 1 .. 10) # _Stefano Spezia_, Dec 23 2018
%t T[n_, m_] = (Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m)^2; Flatten[Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]]
%o (GAP) Flat(List([1..10], n->List([1..n], m->(Binomial(n-1, m-1)*Binomial(n, m-1)/m)^2))); # _Stefano Spezia_, Dec 23 2018
%o (PARI)
%o T(n, m)= (binomial(n-1, m-1)*binomial(n, m-1)/m)^2;
%o tabl(nn) = for(n=1, nn, for(m=1, n, print1(T(n, m), ", ")); print);
%o tabl(10) \\ _Stefano Spezia_, Dec 23 2018
%Y Cf. A001263 (Narayana numbers), A007318.
%Y Cf. A319743 (row sums).
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_, Mar 10 2010
%E Edited by _Stefano Spezia_, Dec 23 2018