cp's OEIS Frontend

A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

%I A357488 #11 Oct 04 2022 08:40:18
%S A357488 1,0,1,2,4,5,9,13,23,34,54,78,120,170,252,358,517,725,1030,1427,1992,
%T A357488 2733,3759,5106,6946,9345,12577,16788,22384,29641,39199,51529,67626,
%U A357488 88307,115083,149332,193383,249456,321134,411998,527472,673233,857539,1089223,1380772
%N A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.
%C A357488 A partition of n is a weakly decreasing sequence of positive integers summing to n.
%C A357488 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%F A357488 a(n) = A357189(2n - 1).
%e A357488 The a(1) = 1 through a(7) = 9 partitions:
%e A357488   (1)  .  (311)  (322)  (333)    (443)    (553)
%e A357488                  (421)  (432)    (542)    (652)
%e A357488                         (531)    (641)    (751)
%e A357488                         (51111)  (52211)  (52222)
%e A357488                                  (62111)  (53311)
%e A357488                                           (62221)
%e A357488                                           (63211)
%e A357488                                           (73111)
%e A357488                                           (7111111)
%t A357488 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357488 Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,1,30,2}]
%Y A357488 For product equal to sum we have A001055, compositions A335405.
%Y A357488 The version for compositions appears to be A222763, odd version of A357182.
%Y A357488 These are the odd-indexed terms of A357189, ranked by A357486.
%Y A357488 These partitions are ranked by the odd-sum portion of A357485.
%Y A357488 Except at the start, alternately adding zeros gives A357487.
%Y A357488 A000041 counts partitions, strict A000009.
%Y A357488 A025047 counts alternating compositions.
%Y A357488 A103919 counts partitions by alternating sum, full triangle A344651.
%Y A357488 A357136 counts compositions by alternating sum, full triangle A097805.
%Y A357488 Cf. A004526, A051159, A114220, A131044, A262046, A262977, A357183, A357184.
%K A357488 nonn
%O A357488 1,4
%A A357488 _Gus Wiseman_, Oct 02 2022
%E A357488 More terms from _Alois P. Heinz_, Oct 04 2022