cp's OEIS Frontend

A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

%I A349061 #10 Dec 25 2021 02:44:35
%S A349061 0,0,0,0,0,0,1,2,4,8,13,21,32,48,67,99,133,185,245,333,432,574,732,
%T A349061 957,1208,1554,1941,2468,3060,3844,4731,5893,7204,8898,10816,13268,
%U A349061 16043,19546,23523,28497,34150,41147,49106,58892,70020,83597,99047,117778,139087
%N A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.
%C A349061 Also the number of non-weakly alternating 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.
%e A349061 The a(6) = 1 through a(10) = 13 partitions:
%e A349061   (321)  (421)   (431)    (432)     (532)
%e A349061          (3211)  (521)    (531)     (541)
%e A349061                  (4211)   (621)     (631)
%e A349061                  (32111)  (3321)    (721)
%e A349061                           (4311)    (4321)
%e A349061                           (5211)    (5311)
%e A349061                           (42111)   (6211)
%e A349061                           (321111)  (32221)
%e A349061                                     (33211)
%e A349061                                     (43111)
%e A349061                                     (52111)
%e A349061                                     (421111)
%e A349061                                     (3211111)
%t A349061 Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]
%Y A349061 The strong version for compositions is A345192, ranked by A345168.
%Y A349061 The version for compositions is A349053, ranked by A349057.
%Y A349061 The complement is counted by A349060.
%Y A349061 These partitions are ranked by A349794.
%Y A349061 The non-strict case is A349796, complement A349795.
%Y A349061 The strong case is A349801.
%Y A349061 A000041 counts integer partitions.
%Y A349061 A001250 counts alternating permutations, complement A348615.
%Y A349061 A003242 counts Carlitz (anti-run) compositions.
%Y A349061 A025047 counts alternating compositions, ranked by A345167.
%Y A349061 A025048 and A025049 count directed alternating compositions.
%Y A349061 A096441 counts weakly alternating 0-appended partitions.
%Y A349061 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349061 A349052 counts weakly alternating compositions.
%Y A349061 A349056 counts weakly alternating permutations of prime indices.
%Y A349061 A349798 counts weakly but not strongly alternating perms of prime indices.
%Y A349061 Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.
%K A349061 nonn
%O A349061 0,8
%A A349061 _Gus Wiseman_, Dec 06 2021