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

Author

Gus Wiseman, Apr 24 2023

Keywords

• nonn

Comments

Even bisection of A362558.

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

Examples

			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).
		

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.

Programs

Mathematica

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