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A332253 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.

%I A332253 #14 Dec 23 2020 01:51:15
%S A332253 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 A332253 0,5,80,411,972,1175,708,169,0,6,134,939,3061,5340,5160,2611,541,0,8,
%U A332253 244,2279,9948,23850,33432,27391,12176,2272,0,10,461,6261,38866,132151,267459,331583,247448,102195,17966
%N A332253 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 A332253 Each element of the k-set must be represented in the multiset partition.
%H A332253 Andrew Howroyd, <a href="/A332253/b332253.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A332253 Triangle begins:
%e A332253   1;
%e A332253   0, 1;
%e A332253   0, 1,   1;
%e A332253   0, 2,   6,   4;
%e A332253   0, 2,   9,  12,    5;
%e A332253   0, 3,  22,  51,   48,   16;
%e A332253   0, 4,  50, 199,  346,  275,   82;
%e A332253   0, 5,  80, 411,  972, 1175,  708,  169;
%e A332253   0, 6, 134, 939, 3061, 5340, 5160, 2611, 541;
%e A332253   ...
%e A332253 The T(3,1) = 2 multiset partitions are:
%e A332253     {{1,1,1}}
%e A332253     {{1},{1,1}}
%e A332253 The T(3,2) = 6 multiset partitions are:
%e A332253     {{1,1,2}}
%e A332253     {{1,2,2}}
%e A332253     {{1},{1,2}}
%e A332253     {{1},{2,2}}
%e A332253     {{2},{1,1}}
%e A332253     {{2},{1,2}}
%e A332253 The T(3,3) = 4 multiset partitions are:
%e A332253     {{1,2,3}}
%e A332253     {{1},{2,3}}
%e A332253     {{2},{1,3}}
%e A332253     {{3},{1,2}}
%o A332253 (PARI)
%o A332253 R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
%o A332253 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 A332253 {my(T=M(8)); for(n=1, #T~, print(T[n, ][1..n]))}
%Y A332253 Column k=1 is A000009.
%Y A332253 Right diagonal is A007837.
%Y A332253 Row sums are A326517.
%K A332253 nonn,tabl
%O A332253 0,8
%A A332253 _Andrew Howroyd_, Feb 08 2020