OFFSET
0,13
COMMENTS
The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..209
FORMULA
EXAMPLE
Array begins:
=============================================================
n\k | 0 1 2 3 4 5
----+--------------------------------------------------------
0 | 1 1 0 0 0 0 ...
1 | 1 1 1 1 1 1 ...
2 | 1 1 6 46 544 7983 ...
3 | 1 3 42 1900 184550 29724388 ...
4 | 1 3 268 73028 57835569 99940181999 ...
5 | 1 5 1239 2448599 16550232235 311353753947045 ...
6 | 1 11 7278 75497242 4388476386528 896320470282357104 ...
...
The A(2,2) = 6 matrices are:
[1 1] [1 0] [1 0] [2 1] [2 0] [1 0]
[1 0] [1 1] [0 1] [0 1] [0 2] [1 2]
[0 1] [0 1] [1 1]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, k<=1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 21 2020
STATUS
approved