A188139 - OEIS

%I #21 Jun 10 2023 08:09:13

%S 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 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 7,5,3,2,1,1,112,55,33,23,15,11,7,5,3,2,1,1

%N Triangle by rows, A027293 * A129372 as infinite lower triangular matrices

%C Row sums = A066897: (1, 2, 5, 8, 15, 24, 39,...), total number of odd parts in all partitions of n.

%C 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 First few rows of the triangle =

%e .

%e 1,

%e 1, 1

%e 3, 1, 1

%e 4, 2, 1, 1

%e 8, 3, 2, 1, 1

%e 11, 6, 3, 2, 1, 1

%e 19, 8, 5, 3, 2, 1, 1

%e 26, 13, 7, 5, 3, 2, 1, 1

%e 41, 18, 12, 7, 5, 3, 2, 1, 1

%e 56, 28, 16, 11, 7, 5, 3, 2, 1, 1

%e 83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1

%e 112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1

%e 160, 74, 47, 31, 22, 15, 11, 7, 5, 3, 2, 1, 1,

%e ...

%t Table[Count[Flatten[IntegerPartitions[n+k]], 2*k], {n,1,15}, {k,1,n}] (* _David Scambler_, May 24 2012 *)

%Y Cf. A027293, A066897, A129372.

%K nonn,tabl

%O 1,4

%A _Gary W. Adamson_, Mar 21 2011

%E a(22) ff. corrected and more terms from _Georg Fischer_, Jun 10 2023