cp's OEIS Frontend

A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

%I A349060 #15 Mar 20 2024 05:11:08
%S A349060 1,1,2,3,5,7,10,13,18,22,29,35,45,53,68,77,98,112,140,157,195,218,270,
%T A349060 298,367,404,495,542,658,721,873,949,1145,1245,1494,1615,1934,2091,
%U A349060 2492,2688,3188,3436,4068,4369,5155,5537,6511,6976,8186,8763,10251,10962
%N A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.
%C A349060 Also the number of 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.
%H A349060 John Tyler Rascoe, <a href="/A349060/b349060.txt">Table of n, a(n) for n = 0..300</a>
%F A349060 G.f.: 1 + Sum_{i>0} (x^i/(1-x^i)) * (1 + Sum_{j>i} (x^j/(1-x^j)) / Product_{k=1..j-i-1} (1-x^(2*(i+k)))). - _John Tyler Rascoe_, Mar 20 2024
%e A349060 The a(1) = 1 through a(7) = 13 partitions:
%e A349060   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A349060        (11)  (21)   (22)    (32)     (33)      (43)
%e A349060              (111)  (31)    (41)     (42)      (52)
%e A349060                     (211)   (221)    (51)      (61)
%e A349060                     (1111)  (311)    (222)     (322)
%e A349060                             (2111)   (411)     (331)
%e A349060                             (11111)  (2211)    (511)
%e A349060                                      (3111)    (2221)
%e A349060                                      (21111)   (4111)
%e A349060                                      (111111)  (22111)
%e A349060                                                (31111)
%e A349060                                                (211111)
%e A349060                                                (1111111)
%t A349060 Table[Length[Select[IntegerPartitions[n], SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]
%o A349060 (PARI)
%o A349060 A_x(N)={my(x='x+O('x^N), g= 1 + sum(i=1, N, (x^i/(1-x^i)) * (1 + sum(j=i+1, N-i, (x^j/((1-x^j))) / prod(k=1, j-i-1, 1-x^(2*(i+k)))))));
%o A349060 Vec(g)}
%o A349060 A_x(52) \\ _John Tyler Rascoe_, Mar 20 2024
%Y A349060 Alternating: A025047, ranked by A345167, also A025048 and A025049.
%Y A349060 The strong case is A065033, ranked by A167171.
%Y A349060 A directed version is A096441.
%Y A349060 Non-alternating: A345192, ranked by A345168.
%Y A349060 Weakly alternating: A349052, also A129852 and A129853.
%Y A349060 Non-weakly alternating: A349053, ranked by A349057.
%Y A349060 A version for ordered factorizations is A349059, strong A348610.
%Y A349060 The complement is counted by A349061, strong A349801.
%Y A349060 These partitions are ranked by the complement of A349794.
%Y A349060 The non-strict case is A349795.
%Y A349060 A000041 counts integer partitions, ordered A011782.
%Y A349060 A001250 counts alternating permutations, complement A348615.
%Y A349060 A344604 counts alternating compositions with twins.
%Y A349060 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349060 Cf. A000070, A002865, A003242, A027383, A114901, A117298, A117989, A274230, A345163, A349056, A349796.
%K A349060 nonn
%O A349060 0,3
%A A349060 _Gus Wiseman_, Dec 06 2021