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A174158 Triangle read by rows: T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.

%I A174158 #23 May 04 2021 09:01:18
%S A174158 1,1,1,1,9,1,1,36,36,1,1,100,400,100,1,1,225,2500,2500,225,1,1,441,
%T A174158 11025,30625,11025,441,1,1,784,38416,240100,240100,38416,784,1,1,1296,
%U A174158 112896,1382976,3111696,1382976,112896,1296,1,1,2025,291600,6350400,28005264,28005264,6350400,291600,2025,1
%N A174158 Triangle read by rows: T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
%H A174158 Stefano Spezia, <a href="/A174158/b174158.txt">First 150 rows of the triangle, flattened</a>
%H A174158 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 A174158 T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.
%F A174158 T(n,m) = A001263(n,m)^2.
%F A174158 T(n,m) = A000290(A007318(n - 1, m - 1)*A007318(n, m - 1)/m). - _Stefano Spezia_, Dec 23 2018
%e A174158 n\m | 1         2         3         4         5         6         7
%e A174158 ----|--------------------------------------------------------------
%e A174158   1 | 1
%e A174158   2 | 1         1
%e A174158   3 | 1         9         1
%e A174158   4 | 1        36        36         1
%e A174158   5 | 1       100       400       100         1
%e A174158   6 | 1       225      2500      2500       225         1
%e A174158   7 | 1       441     11025     30625     11025       441         1
%p A174158 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 A174158 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 A174158 (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 A174158 (PARI)
%o A174158 T(n, m)= (binomial(n-1, m-1)*binomial(n, m-1)/m)^2;
%o A174158 tabl(nn) = for(n=1, nn, for(m=1, n, print1(T(n, m), ", ")); print);
%o A174158 tabl(10) \\ _Stefano Spezia_, Dec 23 2018
%Y A174158 Cf. A001263 (Narayana numbers), A007318.
%Y A174158 Cf. A319743 (row sums).
%K A174158 nonn,tabl
%O A174158 1,5
%A A174158 _Roger L. Bagula_, Mar 10 2010
%E A174158 Edited by _Stefano Spezia_, Dec 23 2018