cp's OEIS Frontend

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

%I A357639 #19 Oct 19 2022 18:09:41
%S A357639 1,0,2,1,6,4,15,13,37,37,86,94,194,223,416,497,867,1056,1746,2159,
%T A357639 3424,4272,6546,8215,12248,15418,22449,28311,40415,50985,71543,90222,
%U A357639 124730,157132,214392,269696,363733,456739,609611,763969,1010203,1263248,1656335,2066552,2688866
%N A357639 Number of reversed integer partitions of 2n whose half-alternating sum is 0.
%C A357639 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 - ...
%H A357639 Alois P. Heinz, <a href="/A357639/b357639.txt">Table of n, a(n) for n = 0..250</a> (first 51 terms from Lucas A. Brown)
%H A357639 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A357639.py">A357639.py</a>.
%e A357639 The a(0) = 1 through a(6) = 15 reversed partitions:
%e A357639   ()  .  (112)   (123)  (134)       (145)      (156)
%e A357639          (1111)         (224)       (235)      (246)
%e A357639                         (2222)      (11233)    (336)
%e A357639                         (11222)     (1111123)  (3333)
%e A357639                         (1111112)              (11244)
%e A357639                         (11111111)             (11334)
%e A357639                                                (12333)
%e A357639                                                (1111134)
%e A357639                                                (1111224)
%e A357639                                                (1112223)
%e A357639                                                (1122222)
%e A357639                                                (11112222)
%e A357639                                                (111111222)
%e A357639                                                (11111111112)
%e A357639                                                (111111111111)
%t A357639 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357639 Table[Length[Select[IntegerPartitions[2n],halfats[Reverse[#]]==0&]],{n,0,15}]
%Y A357639 The non-reverse version is A035363/A035444.
%Y A357639 The non-reverse skew version appears to be A035544/A035594.
%Y A357639 These partitions are ranked by A357631, skew A357632.
%Y A357639 The skew-alternating version is A357640.
%Y A357639 This is the central column of A357704.
%Y A357639 A000041 counts integer partitions (also reversed integer partitions).
%Y A357639 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357639 A344651 counts alternating sum of partitions by length, ordered A097805.
%Y A357639 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357639 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357639 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357639 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357639 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357639 A357637 counts partitions by half-alternating sum, skew A357637.
%Y A357639 Cf. A029862, A053251, A357189, A357487, A357488, A357631, A357634, A357636, A357639, A357641, A357645.
%K A357639 nonn
%O A357639 0,3
%A A357639 _Gus Wiseman_, Oct 11 2022
%E A357639 a(31) onwards from _Lucas A. Brown_, Oct 19 2022