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

1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 29, 35, 45, 53, 68, 77, 98, 112, 140, 157, 195, 218, 270, 298, 367, 404, 495, 542, 658, 721, 873, 949, 1145, 1245, 1494, 1615, 1934, 2091, 2492, 2688, 3188, 3436, 4068, 4369, 5155, 5537, 6511, 6976, 8186, 8763, 10251, 10962

Offset: 0

Author

Gus Wiseman, Dec 06 2021

Keywords

• nonn

Comments

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.

Examples

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..300

Crossrefs

Alternating: A025047, ranked by A345167, also A025048 and A025049.

The strong case is A065033, ranked by A167171.

A directed version is A096441.

Non-alternating: A345192, ranked by A345168.

Weakly alternating: A349052, also A129852 and A129853.

Non-weakly alternating: A349053, ranked by A349057.

A version for ordered factorizations is A349059, strong A348610.

The complement is counted by A349061, strong A349801.

These partitions are ranked by the complement of A349794.

The non-strict case is A349795.

A000041 counts integer partitions, ordered A011782.

A001250 counts alternating permutations, complement A348615.

A344604 counts alternating compositions with twins.

A345170 counts partitions w/ an alternating permutation, ranked by A345172.

Cf. A000070, A002865, A003242, A027383, A114901, A117298, A117989, A274230, A345163, A349056, A349796.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

PARI

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)))))));
Vec(g)}
A_x(52) \\ John Tyler Rascoe, Mar 20 2024

Formula

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