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

1,3

Comments

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

References

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.

Links

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

Formula

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

Example

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.

Mathematica

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 *)

Prog

(Magma)
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
(SageMath)
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

Crossrefs

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

Sequence in context: A323317 A246634 A270508 * A145413 A367459 A279442

Adjacent sequences: A134645 A134646 A134647 * A134649 A134650 A134651

Keyword

nonn

Author

Shanzhen Gao, Nov 05 2007

Extensions

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

Status

approved