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%I #24 May 19 2019 09:30:00
%S 1,1,1,1,2,4,6,6,1,3,9,27,78,210,510,1050,1680,1680,1,4,16,64,256,
%T 1020,4020,15540,58380,210840,722400,2310000,6745200,17417400,
%U 37837800,63063000,63063000,1,5,25,125,625,3125,15620,77980,388220,1923180,9454620
%N T(n,k) is the number of size k ordered submultisets of the regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 1 <= i <= n).
%C A181567 gives the case for unordered submultisets.
%H Alois P. Heinz, <a href="/A234574/b234574.txt">Rows n = 0..26, flattened</a>
%H Thomas Wieder, <a href="/A234574/a234574.txt">Maple program for A234574</a>
%e For n=2 we have the regular multiset L = [1,1,2,2].
%e We get the following ordered submultisets from L:
%e For k=0 1 multiset: []
%e For k=1 2 multisets: [1], [2]
%e For k=2 4 multisets: [1,1], [1,2], [2,1], [2,2]
%e For k=3 6 multisets: [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1]
%e For k=4 6 multisets: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
%e Triangle begins with:
%e 1;
%e 1, 1;
%e 1, 2, 4, 6, 6;
%e 1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680;
%e 1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000;
%e ...
%p # first Maple program: see link above
%p # second Maple program:
%p b:= proc(n, k, i) option remember; `if`(k=0, 1,
%p `if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..n)))
%p end:
%p T:= (n, k)-> b(n, k, n)*k!:
%p seq(seq(T(n, k), k=0..n^2), n=0..5); # _Alois P. Heinz_, Jul 04 2016
%Y Cf. A181567.
%Y Row sums give A274762.
%K nonn,tabf
%O 0,5
%A _Thomas Wieder_, Dec 29 2013
%E More terms from _Alois P. Heinz_, Jul 04 2016
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