A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681

Offset

0,3

Comments

Even bisection of A362558.

a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Links

Table of n, a(n) for n=0..38.

Example

The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Mathematica

Table[Length[Select[IntegerPartitions[2n], !MemberQ[Accumulate[#], n]&]], {n, 0, 15}]

Crossrefs

The version for compositions is A000302, bisection of A213173.

The complement is counted by A322439.

Even bisection of A362558.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with all equal run-sums.

A325347 counts partitions with integer median, complement A307683.

A353836 counts partitions by number of distinct run-sums.

A359893/A359901/A359902 count partitions by median.

Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Sequence in context: A007186 A033304 A091622 * A191315 A184884 A275222

Adjacent sequences: A362048 A362049 A362050 * A362052 A362053 A362054

Keyword

nonn

Author

Gus Wiseman, Apr 24 2023

Status

approved