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Triangle read by rows: T(n,k) is the number of multiset partitions of weight n whose union is a k-set where each part has a different size.
3

%I #14 Dec 23 2020 01:51:15

%S 1,0,1,0,1,1,0,2,6,4,0,2,9,12,5,0,3,22,51,48,16,0,4,50,199,346,275,82,

%T 0,5,80,411,972,1175,708,169,0,6,134,939,3061,5340,5160,2611,541,0,8,

%U 244,2279,9948,23850,33432,27391,12176,2272,0,10,461,6261,38866,132151,267459,331583,247448,102195,17966

%N Triangle read by rows: T(n,k) is the number of multiset partitions of weight n whose union is a k-set where each part has a different size.

%C Each element of the k-set must be represented in the multiset partition.

%H Andrew Howroyd, <a href="/A332253/b332253.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 2, 6, 4;

%e 0, 2, 9, 12, 5;

%e 0, 3, 22, 51, 48, 16;

%e 0, 4, 50, 199, 346, 275, 82;

%e 0, 5, 80, 411, 972, 1175, 708, 169;

%e 0, 6, 134, 939, 3061, 5340, 5160, 2611, 541;

%e ...

%e The T(3,1) = 2 multiset partitions are:

%e {{1,1,1}}

%e {{1},{1,1}}

%e The T(3,2) = 6 multiset partitions are:

%e {{1,1,2}}

%e {{1,2,2}}

%e {{1},{1,2}}

%e {{1},{2,2}}

%e {{2},{1,1}}

%e {{2},{1,2}}

%e The T(3,3) = 4 multiset partitions are:

%e {{1,2,3}}

%e {{1},{2,3}}

%e {{2},{1,3}}

%e {{3},{1,2}}

%o (PARI)

%o R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}

%o M(n)={my(v=vector(n+1, k, R(n, k-1)~)); Mat(vector(n+1, k, k--; sum(i=0, k, (-1)^(k-i)*binomial(k, i)*v[1+i])))}

%o {my(T=M(8)); for(n=1, #T~, print(T[n, ][1..n]))}

%Y Column k=1 is A000009.

%Y Right diagonal is A007837.

%Y Row sums are A326517.

%K nonn,tabl

%O 0,8

%A _Andrew Howroyd_, Feb 08 2020