cp's OEIS Frontend

A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

%I A351593 #9 Mar 13 2022 11:14:19
%S A351593 0,1,1,1,1,2,1,2,2,2,2,4,2,4,3,5,4,6,4,8,6,9,6,12,7,14,10,16,11,20,13,
%T A351593 24,16,28,18,34,21,40,26,46,30,56,34,64,41,75,48,88,54,102,64,118,73,
%U A351593 138,84,159,98,182,112,210,128,242,148,276,168,318
%N A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.
%C A351593 Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.
%e A351593 The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
%e A351593   1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
%e A351593               221     331  332  441  442  443    552  553    554  663
%e A351593                                           551         661    662  771
%e A351593                                           33221       44221       44331
%e A351593                                                                   55221
%t A351593 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]
%Y A351593 The even-length ordered version is A003242, ranked by A351010.
%Y A351593 The opposite version is A053251, even-length A351007, any length A351006.
%Y A351593 This is the odd-length case of A351005, even-length A035457.
%Y A351593 With only equalities we get:
%Y A351593   - opposite any length: A351003
%Y A351593   - opposite odd-length: A000009  (except at 0)
%Y A351593   - opposite even-length: A351012
%Y A351593 - any length: A351004
%Y A351593 - odd-length: A351594
%Y A351593 - even-length: A035363
%Y A351593 Without equalities we get:
%Y A351593   - opposite any length: A122129 (apparently)
%Y A351593   - opposite odd-length: A122130 (apparently)
%Y A351593   - opposite even-length: A351008
%Y A351593 - any length: A122135 (apparently)
%Y A351593 - odd-length: A351595
%Y A351593 - even-length: A122134 (apparently)
%Y A351593 Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.
%K A351593 nonn
%O A351593 0,6
%A A351593 _Gus Wiseman_, Feb 23 2022