This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373270 #23 Jun 02 2024 19:28:56
%S A373270 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,4,3,2,1,1,3,7,6,4,2,1,1,4,8,8,6,4,
%T A373270 2,1,1,4,10,12,10,5,4,2,1,1,5,12,15,13,11,6,4,2,1,1,5,15,21,20,17,11,
%U A373270 6,4,2,1,1,6,16,25,26,21,16,10,6,4,2,1,1,6,20,33,36,34,24,17,11,6,4,2,1,1,7,22,38,46,44,34,25,17,11,6,4,2,1,1,7,25,48,58,56,50,38,24,16,11,6,4,2,1
%N A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.
%H A373270 Alois P. Heinz, <a href="/A373270/b373270.txt">Rows n = 0..200, flattened</a> (first 40 rows from Olivier Gérard)
%e A373270 Array begins:
%e A373270 1,
%e A373270 1, 1,
%e A373270 1, 1, 1,
%e A373270 1, 2, 2, 1,
%e A373270 1, 2, 4, 2, 1,
%e A373270 1, 3, 4, 3, 2, 1,
%e A373270 1, 3, 7, 6, 4, 2, 1,
%e A373270 1, 4, 8, 8, 6, 4, 2, 1,
%e A373270 1, 4, 10, 12, 10, 5, 4, 2, 1,
%e A373270 1, 5, 12, 15, 13, 11, 6, 4, 2, 1,
%e A373270 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,
%e A373270 ...
%e A373270 Example of computation:
%e A373270 T(9,3) = 10 because the partitions of 9 into 3 parts are
%e A373270 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
%e A373270 the number of different multiplicities are
%e A373270 2, 1, 1, 2, 2, 1, 1,
%e A373270 and the sum of these multiplicities is 10.
%t A373270 Flatten@Table[
%t A373270 Plus @@@
%t A373270 Table[Map[Length[Union[Length /@ Split[#]]] &,
%t A373270 IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]
%Y A373270 Cf. A373242, A373269, A373271.
%K A373270 nonn,tabl
%O A373270 1,8
%A A373270 _Olivier Gérard_, May 29 2024