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Triangle read by rows: T(n,k) = max_{c in P(n,n-k+1)} lcm(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.
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%I #12 Apr 20 2021 20:18:27

%S 1,1,2,1,2,3,1,2,3,4,1,2,3,6,5,1,2,3,6,5,6,1,2,3,6,6,12,7,1,2,3,6,6,

%T 12,15,8,1,2,3,6,6,12,15,20,9,1,2,3,6,6,12,15,30,21,10,1,2,3,6,6,12,

%U 15,30,21,30,11,1,2,3,6,6,12,15,30,30,60,35,12

%N Triangle read by rows: T(n,k) = max_{c in P(n,n-k+1)} lcm(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

%C See A181842 for the definition of 'partition'. T(n,k) is also the triangle read by rows: T(n,k) = max_{c in C(n,n-k+1)} lcm(c) where C(n,m) is the set of all m-tuples of positive integers whose elements sum to n where the C(n,k) = A007318(n-1,k-1) are called compositions of n of size k.

%H Andrew Howroyd, <a href="/A181845/b181845.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%e [1] 1

%e [2] 1 2

%e [3] 1 2 3

%e [4] 1 2 3 4

%e [5] 1 2 3 6 5

%e [6] 1 2 3 6 5 6

%e [7] 1 2 3 6 6 12 7

%e [8] 1 2 3 6 6 12 15 8

%e [9] 1 2 3 6 6 12 15 20 9

%p with(combstruct):

%p a181845_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 # alternatively (but slower)

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

%p while not finished(part) do

%p l := nextstruct(part);

%p L := max(L,ilcm(op(l)));

%p od;

%p R := R,L;

%p od;

%p R end:

%o (PARI) Row(n)={my(v=vector(n)); forpart(p=n, my(i=#p); v[i]=max(v[i], lcm(Vec(p)))); Vecrev(v)}

%o { for(n=1, 10, print(Row(n))) } \\ _Andrew Howroyd_, Apr 20 2021

%Y Cf. A181842, A181843, A181844.

%K nonn,tabl

%O 1,3

%A _Peter Luschny_, Dec 07 2010

%E Terms a(56) and beyond from _Andrew Howroyd_, Apr 20 2021