A122934 - OEIS

%I #19 Jan 30 2022 11:41:56

%S 1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,3,2,2,1,1,1,1,3,2,2,1,1,1,3,2,4,2,2,

%T 1,1,1,2,4,2,4,2,2,1,1,1,3,4,5,3,4,2,2,1,1,1,1,3,4,5,3,4,2,2,1,1,1,5,

%U 4,6,5,6,3,4,2,2,1,1,1,1,5,4,6,5,6,3,4,2,2,1,1,1,3,4,7,6,7,6,6,3,4,2,2,1,1

%N Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

%H Seiichi Manyama, <a href="/A122934/b122934.txt">Rows n = 1..140, flattened (Rows n = 1..50 from G. C. Greubel)</a>

%H M. Benoumhani, M. Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS 13 (2010) # 10.3.5

%F T(n,1) = 1. T(n,k+1) = Sum_{d|n, d<n} T(n/d-1,k) = Sum_{d|n, d>1} T(d-1,k).

%e Triangle starts:

%e   1;

%e   1, 1;

%e   1, 1, 1;

%e   1, 2, 1, 1;

%e   1, 1, 2, 1, 1;

%e   1, 3, 2, 2, 1, 1;

%e   ...

%e T(6,3) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.

%t T[_, 1] = 1; T[n_, k_] := T[n, k] = DivisorSum[n, If[#==1, 0, T[#-1, k-1]]& ]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 30 2016 *)

%Y Column k=1..4 give A057427, A032741, A049822, A121895.

%Y Row sums give A003238.

%K easy,nonn,tabl

%O 1,8

%A _Franklin T. Adams-Watters_, Sep 20 2006