%I #16 Jun 09 2021 06:22:38

%S 0,0,0,1,1,3,3,7,7,14,15,27,29,49,54,86,96,146,165,242,275,392,449,

%T 623,716,973,1123,1498,1732,2274,2635,3411,3955,5059,5871,7427,8620,

%U 10801,12536,15572,18065,22267,25821,31602,36617,44533,51560,62338,72105,86716

%N Number of integer partitions of n with reverse-alternating sum < 0.

%C The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

%C Also the number of reversed of integer partitions of n with alternating sum < 0.

%C No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.

%C Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).

%C A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

%e The a(3) = 1 through a(9) = 14 partitions:

%e (21) (31) (32) (42) (43) (53) (54)

%e (41) (51) (52) (62) (63)

%e (2111) (3111) (61) (71) (72)

%e (2221) (3221) (81)

%e (3211) (4211) (3222)

%e (4111) (5111) (3321)

%e (211111) (311111) (4221)

%e (4311)

%e (5211)

%e (6111)

%e (222111)

%e (321111)

%e (411111)

%e (21111111)

%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];

%t Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30}]

%Y The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.

%Y The strict case (for n > 2) is A067659 (odd bisection: A344650).

%Y The Heinz numbers of these partitions are A119899 (complement: A344609).

%Y The bisections are A236914 (odd) and A344743 (even).

%Y The ordered version appears to be A294175 (even bisection: A008549).

%Y The complement is counted by A344607 (even bisection: A344611).

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A027187 counts partitions with alternating sum <= 0, ranked by A028260.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A120452 counts partitions with rev-alternating sum 2 (negative: A344741).

%Y A316524 is the alternating sum of the prime indices of n (reverse: A344616).

%Y A325534/A325535 count separable/inseparable partitions.

%Y A344604 counts wiggly compositions with twins.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y A344618 gives reverse-alternating sums of standard compositions.

%Y Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

%K nonn

%O 0,6

%A _Gus Wiseman_, May 30 2021