OFFSET
0,5
COMMENTS
T(n,k) is the number of 2*n X 2*k {-1,1} matrices with all rows and columns summing to zero.
LINKS
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.
FORMULA
T(n,k) = T(k,n).
EXAMPLE
Array begins:
========================================================================
n\k | 0 1 2 3 4 5 ...
----+------------------------------------------------------------------
0 | 1 1 1 1 1 1 ...
1 | 1 2 6 20 70 252 ...
2 | 1 6 90 1860 44730 1172556 ...
3 | 1 20 1860 297200 60871300 14367744720 ...
4 | 1 70 44730 60871300 116963796250 273957842462220 ...
5 | 1 252 1172556 14367744720 273957842462220 6736218287430460752 ...
...
PROG
(PARI)
T(n, k)={
local(M=Map(Mat([2*k, 1])));
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p, i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
for(r=1, 2*n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[, 2]);
}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Oct 11 2024
STATUS
approved