cp's OEIS Frontend

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

%I A365067 #11 Oct 23 2023 21:43:23
%S A365067 1,1,1,1,1,2,2,1,2,2,2,3,3,2,2,4,3,4,3,5,5,3,4,4,6,5,6,6,5,8,7,5,6,8,
%T A365067 6,10,7,10,9,10,8,12,11,7,10,12,12,10,15,11,14,15,15,16,12,18,15,11,
%U A365067 14,20,18,20,15,22,15,22,21,25,24,24,18,27
%N A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.
%C A365067 The version for all k = 0..n is A113685 (including zeros).
%F A365067 T(n,k) = A000009(k) * A000041((n-k)/2).
%e A365067 Triangle begins:
%e A365067    1
%e A365067    1
%e A365067    1  1
%e A365067    1  2
%e A365067    2  1  2
%e A365067    2  2  3
%e A365067    3  2  2  4
%e A365067    3  4  3  5
%e A365067    5  3  4  4  6
%e A365067    5  6  6  5  8
%e A365067    7  5  6  8  6 10
%e A365067    7 10  9 10  8 12
%e A365067   11  7 10 12 12 10 15
%e A365067   11 14 15 15 16 12 18
%e A365067   15 11 14 20 18 20 15 22
%e A365067   15 22 21 25 24 24 18 27
%e A365067 Row n = 8 counts the following partitions:
%e A365067   (8)     (611)    (431)     (521)      (71)
%e A365067   (62)    (4211)   (41111)   (332)      (53)
%e A365067   (44)    (22211)  (3221)    (32111)    (5111)
%e A365067   (422)            (221111)  (2111111)  (3311)
%e A365067   (2222)                                (311111)
%e A365067                                         (11111111)
%e A365067 Row n = 9 counts the following partitions:
%e A365067   (81)     (63)      (54)       (72)        (9)
%e A365067   (621)    (6111)    (522)      (5211)      (711)
%e A365067   (441)    (432)     (4311)     (3321)      (531)
%e A365067   (4221)   (42111)   (411111)   (321111)    (51111)
%e A365067   (22221)  (3222)    (32211)    (21111111)  (333)
%e A365067            (222111)  (2211111)              (33111)
%e A365067                                             (3111111)
%e A365067                                             (111111111)
%t A365067 Table[Length[Select[IntegerPartitions[n], Total[Select[#,OddQ]]==k&]],{n,0,15},{k,Mod[n,2],n,2}]
%Y A365067 Row sums are A000041.
%Y A365067 The version including all k is A113685, even version A113686.
%Y A365067 Column k = 1 is A119620.
%Y A365067 The even version and the reverse version are both A174713.
%Y A365067 For odd-indexed instead of odd parts we have A346697, even version A346698.
%Y A365067 The corresponding rank statistic is A366528, even version A366531.
%Y A365067 A000009 counts partitions into odd parts, ranks A066208.
%Y A365067 A086543 counts partitions with odd parts, ranks A366322.
%Y A365067 A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A365067 Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.
%K A365067 nonn,tabf
%O A365067 0,6
%A A365067 _Gus Wiseman_, Oct 16 2023