VOOZH about

URL: https://oeis.org/A376935

⇱ A376935 - OEIS

login
A376935
Array read by antidiagonals: T(n,k) is the number of 2*n X 2*k binary matrices with all row sums k and column sums n.
10
1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 20, 90, 20, 1, 1, 70, 1860, 1860, 70, 1, 1, 252, 44730, 297200, 44730, 252, 1, 1, 924, 1172556, 60871300, 60871300, 1172556, 924, 1, 1, 3432, 32496156, 14367744720, 116963796250, 14367744720, 32496156, 3432, 1, 1, 12870, 936369720, 3718394156400, 273957842462220, 273957842462220, 3718394156400, 936369720, 12870, 1
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
Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
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
Main diagonal is A058527.
Cf. A008300, A195644, A333901, A334549, A377007 (up to permutations of rows and columns).
Sequence in context: A172373 A174411 A322620 * A155795 A009963 A008300
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Oct 11 2024
STATUS
approved