A357640 Number of reversed integer partitions of 2n whose skew-alternating sum is 0.

1, 1, 2, 3, 6, 9, 16, 24, 40, 59, 93, 136, 208, 299, 445, 632, 921, 1292, 1848, 2563, 3610, 4954, 6881, 9353, 12835, 17290, 23469, 31357, 42150, 55889, 74463, 98038, 129573, 169476, 222339, 289029, 376618, 486773, 630313, 810285, 1043123, 1334174

Offset: 0

Author

Gus Wiseman, Oct 11 2022

Keywords

• nonn

Comments

We define the skew-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(5) = 9 partitions:
  ()  (11)  (22)    (33)      (44)        (55)
            (1111)  (2211)    (2222)      (3322)
                    (111111)  (3221)      (4321)
                              (3311)      (4411)
                              (221111)    (222211)
                              (11111111)  (322111)
                                          (331111)
                                          (22111111)
                                          (1111111111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from Lucas A. Brown)
• Lucas A. Brown, A357640.py.

Crossrefs

The non-reverse half-alternating version is A035363/A035444.

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

These partitions are ranked by A357632, half A357631.

The half-alternating version is A357639.

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 A357638.

Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357636, A357641, A357645, A357704.

Programs

Mathematica

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

Extensions

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