OFFSET
0,13
COMMENTS
The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..209
EXAMPLE
Array begins:
===================================================================
n\k | 0 1 2 3 4 5 6
----+--------------------------------------------------------------
0 | 1 1 0 0 0 0 0 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 1 4 27 266 3599 62941 ...
3 | 1 2 15 317 12586 803764 75603729 ...
4 | 1 2 44 2763 390399 103678954 46278915417 ...
5 | 1 3 120 21006 10074052 10679934500 21806685647346 ...
6 | 1 4 319 147296 232165926 956594630508 8717423133548684 ...
7 | 1 5 804 967829 4903530137 76812482919237 ...
...
The A(2,2) = 4 matrices are:
[2 1] [2 0] [1 2] [1 1]
[0 1] [0 2] [1 0] [1 0]
[0 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)*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)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 10 2020
STATUS
approved