cp's OEIS Frontend

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

%I A357640 #16 Oct 19 2022 18:49:38
%S A357640 1,1,2,3,6,9,16,24,40,59,93,136,208,299,445,632,921,1292,1848,2563,
%T A357640 3610,4954,6881,9353,12835,17290,23469,31357,42150,55889,74463,98038,
%U A357640 129573,169476,222339,289029,376618,486773,630313,810285,1043123,1334174
%N A357640 Number of reversed integer partitions of 2n whose skew-alternating sum is 0.
%C A357640 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 A357640 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 A357640 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A357640.py">A357640.py</a>.
%e A357640 The a(0) = 1 through a(5) = 9 partitions:
%e A357640   ()  (11)  (22)    (33)      (44)        (55)
%e A357640             (1111)  (2211)    (2222)      (3322)
%e A357640                     (111111)  (3221)      (4321)
%e A357640                               (3311)      (4411)
%e A357640                               (221111)    (222211)
%e A357640                               (11111111)  (322111)
%e A357640                                           (331111)
%e A357640                                           (22111111)
%e A357640                                           (1111111111)
%t A357640 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357640 Table[Length[Select[IntegerPartitions[2n],skats[Reverse[#]]==0&]],{n,0,15}]
%Y A357640 The non-reverse half-alternating version is A035363/A035444.
%Y A357640 The non-reverse version appears to be A035544/A035594.
%Y A357640 These partitions are ranked by A357632, half A357631.
%Y A357640 The half-alternating version is A357639.
%Y A357640 A000041 counts integer partitions (also reversed integer partitions).
%Y A357640 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357640 A344651 counts alternating sum of partitions by length, ordered A097805.
%Y A357640 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357640 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357640 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357640 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357640 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357640 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357640 Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357636, A357641, A357645, A357704.
%K A357640 nonn
%O A357640 0,3
%A A357640 _Gus Wiseman_, Oct 11 2022
%E A357640 a(31) onwards from _Lucas A. Brown_, Oct 19 2022