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%I #21 Oct 21 2023 04:54:35
%S 0,1,90,44730,56586600,154700988750,807998767676100,
%T 7373018003758407000,109829050417159537464000,
%U 2532230252503738514963235000,86574740102712303011539719750000,4237239732072431006302896746240010000
%N Number of 2n X n (0,1)-matrices with row sums 2 and column sums 4.
%C t(m,n) in the formula gives the number of (0,1)-matrices of size m*n with row sum 4 and column sum 2. a(n) in the formula gives the number of (0,1)-matrices of size n*(2n) with row sum 4 and column sum 2. - _Shanzhen Gao_, Feb 16 2010
%D Gao, Shanzhen, and Matheis, Kenneth, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
%H R. H. Hardin, <a href="/A134648/b134648.txt">Table of n, a(n) for n = 1..49</a>
%F a(n) = (2*n)!*t(n,n)/n!, where t(m, n) = (1/24^m)*Sum_{j=0..m} Sum_{k=0..m-j} ( (-1)^(m-j-k)*3^j*6^(m-j-k)*m!*n!*(4*k+2*(m-j-k))! )/( j!*k!*(m-j-k)!*(2*k+(m-j-k))!*2^(2*k+(m-j-k)) ).
%F a(n) = (1/24^n)*Sum_{j=0..n} Sum_{k=0..n-j} ((-1)^(n-j-k)*3^j*6^(n-j-k)*n!(2n)!(2n-2j+2k)!/(j!k!(n-j-k)!(n-j+k)!*2^(n-j+k))). - _Shanzhen Gao_, Feb 16 2010
%F a(n) ~ sqrt(Pi) * 2^(3*n + 3/2) * n^(4*n + 1/2) / (3^n * exp(4*n + 3/2)). - _Vaclav Kotesovec_, Oct 21 2023
%e Number of 4 X 2 (0,1)-matrices: 1;
%e Number of 6 X 3 (0,1)-matrices: 90;
%e Number of 8 X 4 (0,1)-matrices: 44730;
%e Number of 10 X 5 (0,1)-matrices: 5658660.
%t t[m_, n_]:= t[m, n]= ((-1)^m*n!/8^m)*Sum[Binomial[m,k]*Binomial[m-k,j]*Binomial[2*m+2*k-2*j,m+k-j]*(m+k-j)!*(-1)^(j+k)/(12)^k, {j,0, m}, {k,0,m-j}];
%t A134648[n_]:= (2*n)!*t[n,n]/n!;
%t Table[A134648[n], {n,30}] (* _G. C. Greubel_, Oct 13 2023 *)
%o (Magma)
%o B:=Binomial; F:=Factorial;
%o f:= func< m,n,k,j | B(m, k)*B(m-k, j)*B(2*m+2*k-2*j, m+k-j)*F(m+k-j) >;
%o t:= func< m,n | ((-1)^m*F(n)/8^m)*(&+[(&+[f(m,n,k,j)*(-1)^(j+k)/(12)^k: k in [0..m-j]]): j in [0..m]]) >;
%o A134648:= func< n | F(2*n)*t(n,n)/F(n) >;
%o [A134648(n): n in [1..30]]; // _G. C. Greubel_, Oct 13 2023
%o (SageMath)
%o b=binomial; F=factorial;
%o def f(m,n,k,j): return b(m, k)*b(m-k, j)*b(2*m+2*k-2*j, m+k-j)*F(m+k-j)
%o def t(m,n): return ((-1)^m*F(n)/8^m)*sum(sum(f(m,n,k,j)*(-1)^(j+k)/(12)^k for k in range(m-j+1)) for j in range(m+1))
%o def A134648(n): return F(2*n)*t(n,n)/F(n)
%o [A134648(n) for n in range(1,31)] # _G. C. Greubel_, Oct 13 2023
%Y Cf. A000681, A000986, A132202, A134645, A134646, A139670, etc.
%K nonn
%O 1,3
%A _Shanzhen Gao_, Nov 05 2007
%E a(7) onwards from _R. H. Hardin_, Oct 18 2009
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