A357640 - OEIS

A357640

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

%I #16 Oct 19 2022 18:49:38

%S 1,1,2,3,6,9,16,24,40,59,93,136,208,299,445,632,921,1292,1848,2563,

%T 3610,4954,6881,9353,12835,17290,23469,31357,42150,55889,74463,98038,

%U 129573,169476,222339,289029,376618,486773,630313,810285,1043123,1334174

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

%C 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 + ...

%H Alois P. Heinz, <a href="/A357640/b357640.txt">Table of n, a(n) for n = 0..250</a> (first 51 terms from Lucas A. Brown)

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A357640.py">A357640.py</a>.

%e The a(0) = 1 through a(5) = 9 partitions:

%e () (11) (22) (33) (44) (55)

%e (1111) (2211) (2222) (3322)

%e (111111) (3221) (4321)

%e (3311) (4411)

%e (221111) (222211)

%e (11111111) (322111)

%e (331111)

%e (22111111)

%e (1111111111)

%t skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];

%t Table[Length[Select[IntegerPartitions[2n],skats[Reverse[#]]==0&]],{n,0,15}]

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

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

%Y These partitions are ranked by A357632, half A357631.

%Y The half-alternating version is A357639.

%Y A000041 counts integer partitions (also reversed integer partitions).

%Y A316524 gives alternating sum of prime indices, reverse A344616.

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

%Y A351005 = alternately equal and unequal partitions, compositions A357643.

%Y A351006 = alternately unequal and equal partitions, compositions A357644.

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

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

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

%Y A357637 counts partitions by half-alternating sum, skew A357638.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 11 2022

%E a(31) onwards from _Lucas A. Brown_, Oct 19 2022