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A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

%I A122934 #22 Jun 22 2025 21:32:48
%S A122934 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 A122934 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 A122934 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 A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.
%H A122934 Seiichi Manyama, <a href="/A122934/b122934.txt">Rows n = 1..140, flattened (Rows n = 1..50 from G. C. Greubel)</a>
%H A122934 M. Benoumhani and 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 A122934 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 A122934 Triangle starts:
%e A122934   1;
%e A122934   1, 1;
%e A122934   1, 1, 1;
%e A122934   1, 2, 1, 1;
%e A122934   1, 1, 2, 1, 1;
%e A122934   1, 3, 2, 2, 1, 1;
%e A122934   ...
%e A122934 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 A122934 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 A122934 Column k=1..4 give A057427, A032741, A049822, A121895.
%Y A122934 Row sums give A003238.
%K A122934 easy,nonn,tabl
%O A122934 1,8
%A A122934 _Franklin T. Adams-Watters_, Sep 20 2006