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