A134646 -id:A134646 - cp's OEIS frontend

A134648 Number of 2n X n (0,1)-matrices with row sums 2 and column sums 4.

0, 1, 90, 44730, 56586600, 154700988750, 807998767676100, 7373018003758407000, 109829050417159537464000, 2532230252503738514963235000, 86574740102712303011539719750000, 4237239732072431006302896746240010000

Offset: 1

Shanzhen Gao, Nov 05 2007

nonn

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

			Number of  4 X 2 (0,1)-matrices:       1;
Number of  6 X 3 (0,1)-matrices:      90;
Number of  8 X 4 (0,1)-matrices:   44730;
Number of 10 X 5 (0,1)-matrices: 5658660.

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.

R. H. Hardin, Table of n, a(n) for n = 1..49

Cf. A000681, A000986, A132202, A134645, A134646, A139670, etc.

B:=Binomial; F:=Factorial;
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) >;
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]]) >;
A134648:= func< n | F(2*n)*t(n,n)/F(n) >;
[A134648(n): n in [1..30]]; // G. C. Greubel, Oct 13 2023

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}];
A134648[n_]:= (2*n)!*t[n,n]/n!;
Table[A134648[n], {n,30}] (* G. C. Greubel, Oct 13 2023 *)

b=binomial; F=factorial;
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)
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))
def A134648(n): return F(2*n)*t(n,n)/F(n)
[A134648(n) for n in range(1,31)] # G. C. Greubel, Oct 13 2023

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)) ).
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
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

a(7) onwards from R. H. Hardin, Oct 18 2009

A134645 Number of 2n X 3n (0,1,2)-matrices with every row sum 3 and column sum 2.

Original entry on oeis.org

7, 16260, 747558000, 250071339672000, 369820640830881240000, 1796185853884657144990080000, 23511842995969107700302647865600000, 720289186703359375552628986978410240000000, 46455761324619133018320834819622638940550400000000, 5809177204262302555518772962193269714031251010176000000000

Offset: 1

Shanzhen Gao, Nov 05 2007

nonn

			a(1) = 7:
111 210 (6 ways)
111 012

Zhonghua Tan, Shanzhen Gao, Kenneth Mathies, Joshua Fallon, Counting (0,1,2)-Matrices, Congressus Numeratium, December 2008.

Cf. A000681, A134646.

f:=proc(m,n) 6^(-m)*add( (3^i*m!*n!*(2*n-2*i)!)/ (i!*(m-i)!*(n-i)!*2^(n-i)), i=0..m); end;

Table[(3*n)! * (2*n)! / 288^n * Sum[(6*n - 2*i)! * 6^i / (i! * (3*n - i)! * (2*n - i)!), {i, 0, 2*n}], {n, 1, 15}] (* Vaclav Kotesovec, Oct 21 2023 *)
Table[(2/9)^n * (3*n)! * ((6*n - 1)/2)! * Hypergeometric1F1[-2*n, 1/2 - 3*n, 3/2] / Sqrt[Pi], {n, 1, 15}] (* Vaclav Kotesovec, Oct 21 2023 *)

Let t(m,n)=6^{-m} sum_{i=0}^{m}frac{3^{i}m!n!(2n-2i)!}{i!(m-i)!(n-i)!2^{n-i}}; then a(n) = t(2n,3n).
a(n) = (3n)!(2n)!288^(-n) * Sum_{i=0..2n} (6n-2i)!6^i/(i!(3n-i)!(2n-i)!). - Shanzhen Gao, Mar 02 2010
a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n-1). - Vaclav Kotesovec, Oct 21 2023

Corrected, edited and extended with Maple program by R. H. Hardin and N. J. A. Sloane, Oct 18 2009

cp's OEIS Frontend

A134648 Number of 2n X n (0,1)-matrices with row sums 2 and column sums 4.

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