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%I A181842 #8 Jul 26 2013 02:55:37
%S A181842 1,1,2,1,2,3,1,2,5,4,1,2,5,10,5,1,2,5,12,12,6,1,2,5,12,18,28,7,1,2,5,
%T A181842 12,20,38,32,8,1,2,5,12,20,44,57,48,9,1,2,5,12,20,46,67,100,55,10
%N A181842 Triangle read by rows: T(n,k) = Sum_{c in partition(n,n-k+1)} lcm(c)
%C A181842 In A181842 through A181854 the following terminology is used.
%C A181842 Let n, k be positive integers.
%C A181842 * Partition: A (n,k)-partition is the set of all k-sets of
%C A181842 positive integers whose elements sum to n.
%C A181842 - The cardinality of a (n,k)-partition: A008284(n,k).
%C A181842 - Maple: (n,k) -> combstruct[count](Partition(n),size=k).
%C A181842 - The (6,2)-partition is {{1,5},{2,4},{3,3}}.
%C A181842 * Composition: A (n,k)-composition is the set of all k-tuples of positive integers whose elements sum to n.
%C A181842 - The cardinality of a (n,k)-composition: A007318(n-1,k-1).
%C A181842 - Maple: (n,k) -> combstruct[count](Composition(n),size=k).
%C A181842 - The (6,2)-composition is {<5,1>,<4,2>,<3,3>,<2,4>,<1,5>}.
%C A181842 * Combination: A (n,k)-combination is the set of all k-subsets
%C A181842 of {1,2,..,n}.
%C A181842 - The cardinality of a (n,k)-combination: A007318(n,k).
%C A181842 - Maple: (n,k) -> combstruct[count](Combination(n),size=k).
%C A181842 - The (4,2)-combination is {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}.
%e A181842 [1]   1
%e A181842 [2]   1   2
%e A181842 [3]   1   2   3
%e A181842 [4]   1   2   5   4
%e A181842 [5]   1   2   5   10   5
%e A181842 [6]   1   2   5   12   12   6
%e A181842 [7]   1   2   5   12   18   28   7
%p A181842 with(combstruct):
%p A181842 a181842_row := proc(n) local k,L,l,R,part;
%p A181842 R := NULL;
%p A181842 for k from 1 to n do
%p A181842    L := 0;
%p A181842    part := iterstructs(Partition(n),size=n-k+1):
%p A181842    while not finished(part) do
%p A181842       l := nextstruct(part);
%p A181842       L := L + ilcm(op(l));
%p A181842    od;
%p A181842    R := R,L;
%p A181842 od;
%p A181842 R end:
%t A181842 t[n_, k_] := LCM @@@ IntegerPartitions[n, {n - k + 1}] // Total; Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 26 2013 *)
%Y A181842 Cf. A181843, A181844.
%K A181842 nonn,tabl
%O A181842 1,3
%A A181842 _Peter Luschny_, Dec 07 2010

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