cp's OEIS Frontend

A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

%I A344743 #10 Jun 12 2021 06:04:57
%S A344743 0,0,1,3,7,15,29,54,96,165,275,449,716,1123,1732,2635,3955,5871,8620,
%T A344743 12536,18065,25821,36617,51560,72105,100204,138417,190134,259772,
%U A344743 353134,477734,643354,862604,1151773,1531738,2029305,2678650,3523378,4618835,6035240,7861292
%N A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.
%C A344743 Conjecture: a(n) >= A236914.
%C A344743 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.
%C A344743 The alternating sum of a partition is never < 0, so the non-reverse version is A000004.
%F A344743 a(n) = A058696(n) - A344611(n).
%F A344743 a(n) = sum of left half of even-indexed rows of A344612.
%e A344743 The a(2) = 1 through a(5) = 15 partitions:
%e A344743   (31)  (42)    (53)      (64)
%e A344743         (51)    (62)      (73)
%e A344743         (3111)  (71)      (82)
%e A344743                 (3221)    (91)
%e A344743                 (4211)    (3331)
%e A344743                 (5111)    (4222)
%e A344743                 (311111)  (4321)
%e A344743                           (5221)
%e A344743                           (5311)
%e A344743                           (6211)
%e A344743                           (7111)
%e A344743                           (322111)
%e A344743                           (421111)
%e A344743                           (511111)
%e A344743                           (31111111)
%t A344743 sats[y_] := Sum[(-1)^(i - Length[y])*y[[i]], {i, Length[y]}];
%t A344743 Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30,2}]
%Y A344743 The ordered version (compositions not partitions) appears to be A008549.
%Y A344743 The Heinz numbers are A119899 /\ A300061.
%Y A344743 Even bisection of A344608.
%Y A344743 The complementary partitions of 2n are counted by A344611.
%Y A344743 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344743 A001523 counts unimodal compositions (partial sums: A174439).
%Y A344743 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344743 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A344743 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A344743 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344743 A325534/A325535 count separable/inseparable partitions.
%Y A344743 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344743 Cf. A000070, A000097, A006330, A027187, A239830, A343941, A344604, A344607, A344650, A344651, A344654.
%K A344743 nonn
%O A344743 0,4
%A A344743 _Gus Wiseman_, Jun 09 2021
%E A344743 More terms from _Bert Dobbelaere_, Jun 12 2021