A357639 Number of reversed integer partitions of 2n whose half-alternating sum is 0.

1, 0, 2, 1, 6, 4, 15, 13, 37, 37, 86, 94, 194, 223, 416, 497, 867, 1056, 1746, 2159, 3424, 4272, 6546, 8215, 12248, 15418, 22449, 28311, 40415, 50985, 71543, 90222, 124730, 157132, 214392, 269696, 363733, 456739, 609611, 763969, 1010203, 1263248, 1656335, 2066552, 2688866

Offset: 0

Author

Gus Wiseman, Oct 11 2022

Keywords

• nonn

Comments

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

Examples

			The a(0) = 1 through a(6) = 15 reversed partitions:
  ()  .  (112)   (123)  (134)       (145)      (156)
         (1111)         (224)       (235)      (246)
                        (2222)      (11233)    (336)
                        (11222)     (1111123)  (3333)
                        (1111112)              (11244)
                        (11111111)             (11334)
                                               (12333)
                                               (1111134)
                                               (1111224)
                                               (1112223)
                                               (1122222)
                                               (11112222)
                                               (111111222)
                                               (11111111112)
                                               (111111111111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from Lucas A. Brown)
• Lucas A. Brown, A357639.py.

Crossrefs

The non-reverse version is A035363/A035444.

The non-reverse skew version appears to be A035544/A035594.

These partitions are ranked by A357631, skew A357632.

The skew-alternating version is A357640.

This is the central column of A357704.

A000041 counts integer partitions (also reversed integer partitions).

A316524 gives alternating sum of prime indices, reverse A344616.

A344651 counts alternating sum of partitions by length, ordered A097805.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew A357634.

A357637 counts partitions by half-alternating sum, skew A357637.

Cf. A029862, A053251, A357189, A357487, A357488, A357631, A357634, A357636, A357639, A357641, A357645.

Programs

Mathematica

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[2n],halfats[Reverse[#]]==0&]],{n,0,15}]

Extensions

a(31) onwards from Lucas A. Brown, Oct 19 2022