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 A188139 #23 Jul 22 2025 11:00:43
%S A188139 1,1,1,3,1,1,4,2,1,1,8,3,2,1,1,11,6,3,2,1,1,19,8,5,3,2,1,1,26,13,7,5,
%T A188139 3,2,1,1,41,18,12,7,5,3,2,1,1,56,28,16,11,7,5,3,2,1,1,83,38,24,15,11,
%U A188139 7,5,3,2,1,1,112,55,33,23,15,11,7,5,3,2,1,1
%N A188139 Triangle by rows, A027293 * A129372 as infinite lower triangular matrices.
%C A188139 Row sums = A066897: (1, 2, 5, 8, 15, 24, 39,...), total number of odd parts in all partitions of n.
%C A188139 Apparently T(n,k) is the number of (2*k)'s in all the partitions of (n+k), k>=1, e.g. T(7,3) = number of 6's in partitions of 10 = A024790(10). [_David Scambler_, May 24 2012]
%e A188139 First few rows of the triangle =
%e A188139 .
%e A188139 1,
%e A188139 1, 1
%e A188139 3, 1, 1
%e A188139 4, 2, 1, 1
%e A188139 8, 3, 2, 1, 1
%e A188139 11, 6, 3, 2, 1, 1
%e A188139 19, 8, 5, 3, 2, 1, 1
%e A188139 26, 13, 7, 5, 3, 2, 1, 1
%e A188139 41, 18, 12, 7, 5, 3, 2, 1, 1
%e A188139 56, 28, 16, 11, 7, 5, 3, 2, 1, 1
%e A188139 83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1
%e A188139 112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1
%e A188139 160, 74, 47, 31, 22, 15, 11, 7, 5, 3, 2, 1, 1,
%e A188139 ...
%t A188139 Table[Count[Flatten[IntegerPartitions[n+k]], 2*k], {n,1,15}, {k,1,n}] (* _David Scambler_, May 24 2012 *)
%Y A188139 Cf. A027293, A066897, A129372.
%K A188139 nonn,tabl
%O A188139 1,4
%A A188139 _Gary W. Adamson_, Mar 21 2011
%E A188139 a(22) ff. corrected and more terms from _Georg Fischer_, Jun 10 2023