A181842 - OEIS

%I #8 Jul 26 2013 02:55:37

%S 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 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 Triangle read by rows: T(n,k) = Sum_{c in partition(n,n-k+1)} lcm(c)

%C In A181842 through A181854 the following terminology is used.

%C Let n, k be positive integers.

%C * Partition: A (n,k)-partition is the set of all k-sets of

%C positive integers whose elements sum to n.

%C - The cardinality of a (n,k)-partition: A008284(n,k).

%C - Maple: (n,k) -> combstruct[count](Partition(n),size=k).

%C - The (6,2)-partition is {{1,5},{2,4},{3,3}}.

%C * Composition: A (n,k)-composition is the set of all k-tuples of positive integers whose elements sum to n.

%C - The cardinality of a (n,k)-composition: A007318(n-1,k-1).

%C - Maple: (n,k) -> combstruct[count](Composition(n),size=k).

%C - The (6,2)-composition is {<5,1>,<4,2>,<3,3>,<2,4>,<1,5>}.

%C * Combination: A (n,k)-combination is the set of all k-subsets

%C of {1,2,..,n}.

%C - The cardinality of a (n,k)-combination: A007318(n,k).

%C - Maple: (n,k) -> combstruct[count](Combination(n),size=k).

%C - The (4,2)-combination is {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}.

%e [1]   1

%e [2]   1   2

%e [3]   1   2   3

%e [4]   1   2   5   4

%e [5]   1   2   5   10   5

%e [6]   1   2   5   12   12   6

%e [7]   1   2   5   12   18   28   7

%p with(combstruct):

%p a181842_row := proc(n) local k,L,l,R,part;

%p R := NULL;

%p for k from 1 to n do

%p    L := 0;

%p    part := iterstructs(Partition(n),size=n-k+1):

%p    while not finished(part) do

%p       l := nextstruct(part);

%p       L := L + ilcm(op(l));

%p    od;

%p    R := R,L;

%p od;

%p R end:

%t 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 Cf. A181843, A181844.

%K nonn,tabl

%O 1,3

%A _Peter Luschny_, Dec 07 2010