OFFSET
0,8
COMMENTS
The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices 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 1 1 1 1 ...
1 | 1 1 2 4 8 16 ...
2 | 1 1 7 59 701 10460 ...
3 | 1 3 45 1987 190379 30474159 ...
4 | 1 3 271 73567 58055460 100171963518 ...
5 | 1 5 1244 2451082 16557581754 311419969572540 ...
6 | 1 11 7289 75511809 4388702900099 ...
...
The A(2,2) = 7 matrices are:
[1 1] [1 0] [1 0] [2 1] [2 0] [1 0] [2 2]
[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 - 1, 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, 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