cp's OEIS Frontend

A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

%I A351595 #6 Mar 13 2022 11:14:25
%S A351595 0,1,1,1,1,2,2,3,4,5,6,9,10,13,16,20,24,30,35,44,52,63,74,90,105,126,
%T A351595 148,175,204,242,280,330,382,446,515,600,690,800,919,1060,1214,1398,
%U A351595 1595,1830,2086,2384,2711,3092,3506,3988,4516,5122,5788,6552,7388,8345
%N A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.
%e A351595 The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
%e A351595   1   2   3   4   5     6     7     8     9     A     B       C
%e A351595                   221   321   331   332   432   442   443     543
%e A351595                               421   431   441   532   542     552
%e A351595                                     521   531   541   551     642
%e A351595                                           621   631   632     651
%e A351595                                                 721   641     732
%e A351595                                                       731     741
%e A351595                                                       821     831
%e A351595                                                       33221   921
%e A351595                                                               43221
%t A351595 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]
%Y A351595 The ordered version (compositions) is A000213 shifted right once.
%Y A351595 All odd-length partitions are counted by A027193.
%Y A351595 The opposite appears to be A122130, even-length A351008, any length A122129.
%Y A351595 This appears to be the odd-length case of A122135, even-length A122134.
%Y A351595 The case that is constant at odd indices:
%Y A351595 - any length: A351005
%Y A351595 - odd length: A351593
%Y A351595 - even length: A035457
%Y A351595 - opposite any length: A351006
%Y A351595 - opposite odd length: A053251
%Y A351595 - opposite even length: A351007
%Y A351595 For equality instead of inequality:
%Y A351595 - any length: A351003
%Y A351595 - odd-length: A000009 (except at 0)
%Y A351595 - even-length: A351012
%Y A351595 - opposite any length: A351004
%Y A351595 - opposite odd-length: A351594
%Y A351595 - opposite even-length: A035363
%Y A351595 Cf. A000041, A000070, A003242, A027383, A236559, A236914, A350842.
%K A351595 nonn
%O A351595 0,6
%A A351595 _Gus Wiseman_, Feb 25 2022