cp's OEIS Frontend

A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

%I A349795 #15 Dec 26 2021 06:14:06
%S A349795 0,0,1,1,3,4,7,9,14,17,24,29,39,46,61,69,90,103,131,147,185,207,259,
%T A349795 286,355,391,482,528,644,706,858,933,1129,1228,1477,1597,1916,2072,
%U A349795 2473,2668,3168,3415,4047,4347,5133,5514,6488,6952,8162,8738,10226,10936
%N A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.
%C A349795 Also the number of weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed. Equivalently, these are partitions that are weakly alternating but not strongly alternating.
%F A349795 a(n > 0) = A349060(n) - A065033(n) = A349060(n) - floor(n/2).
%F A349795 a(n) = A047967(n) - A349796(n).
%e A349795 The a(2) = 1 through a(8) = 14 partitions:
%e A349795   (11)  (111)  (22)    (221)    (33)      (322)      (44)
%e A349795                (211)   (311)    (222)     (331)      (332)
%e A349795                (1111)  (2111)   (411)     (511)      (422)
%e A349795                        (11111)  (2211)    (2221)     (611)
%e A349795                                 (3111)    (4111)     (2222)
%e A349795                                 (21111)   (22111)    (3221)
%e A349795                                 (111111)  (31111)    (3311)
%e A349795                                           (211111)   (5111)
%e A349795                                           (1111111)  (22211)
%e A349795                                                      (41111)
%e A349795                                                      (221111)
%e A349795                                                      (311111)
%e A349795                                                      (2111111)
%e A349795                                                      (11111111)
%t A349795 Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#&&(SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}])&]],{n,0,30}]
%Y A349795 This is the restriction of A349060 to non-strict partitions.
%Y A349795 The complement in non-strict partitions is A349796.
%Y A349795 Permutations of prime factors of this type are counted by A349798.
%Y A349795 The ordered version (compositions) is A349800, ranked by A349799.
%Y A349795 These partitions are ranked by A350137.
%Y A349795 A000041 counts integer partitions, non-strict A047967.
%Y A349795 A001250 counts alternating permutations, complement A348615.
%Y A349795 A025047 counts alternating compositions, also A025048 and A025049.
%Y A349795 A096441 counts weakly alternating 0-appended partitions.
%Y A349795 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349795 A349053 counts non-weakly alternating compositions, complement A349052.
%Y A349795 A349061 counts non-weakly alternating partitions, ranked by A349794.
%Y A349795 A349801 counts non-alternating partitions.
%Y A349795 Cf. A000070, A003242, A027383, A065033, A117298, A129852, A274230, A344614, A344615, A345165, A345167, A347548, A349056, A349057.
%K A349795 nonn
%O A349795 0,5
%A A349795 _Gus Wiseman_, Dec 06 2021