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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 *)

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_Peter Luschny (peter(AT)luschny.de), _, Dec 07 2010

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In A181842 through A181854 the following terminology is used.

Let n, k be positive integers.

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

positive integers whose elements sum to n.

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

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

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

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

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

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

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

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

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

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

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

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

allocated for Peter LuschnyTriangle read by rows: T(n,k) = Sum_{c in partition(n,n-k+1)} lcm(c)

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, 12, 20, 38, 32, 8, 1, 2, 5, 12, 20, 44, 57, 48, 9, 1, 2, 5, 12, 20, 46, 67, 100, 55, 10

1,3

[1] 1

[2] 1 2

[3] 1 2 3

[4] 1 2 5 4

[5] 1 2 5 10 5

[6] 1 2 5 12 12 6

[7] 1 2 5 12 18 28 7

with(combstruct):

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

R := NULL;

for k from 1 to n do

L := 0;

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

while not finished(part) do

l := nextstruct(part);

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

od;

R := R, L;

od;

R end:

Cf. A181843, A181844.

allocated

nonn,tabl

Peter Luschny (peter(AT)luschny.de), Dec 07 2010

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allocated for Peter Luschny

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