A351007 - OEIS

A351007

Number of even-length integer partitions of n into parts that are alternately unequal and equal.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807

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OFFSET

0,6

COMMENTS

These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

LINKS

Table of n, a(n) for n=0..60.

EXAMPLE

The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):

21 31 32 42 43 53 54 64 65 75 76 86 87

41 51 52 62 63 73 74 84 85 95 96

61 71 72 82 83 93 94 A4 A5

3221 81 91 92 A2 A3 B3 B4

4221 5221 A1 B1 B2 C2 C3

4331 4332 C1 D1 D2

6221 5331 5332 5441 E1

7221 6331 6332 5442

8221 7331 6441

9221 7332

8331

A221

433221

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]

CROSSREFS

The alternately equal and unequal version is A035457, any length A351005.

This is the even-length case of A351006, odd-length A053251.

Without equalities we have A351008, any length A122129, opposite A122135.

Without inequalities we have A351012, any length A351003, opposite A351004.

Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

Sequence in context: A112341 A242774 A183003 * A307779 A165684 A342519

Adjacent sequences: A351004 A351005 A351006 * A351008 A351009 A351010

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2022

STATUS

approved