A344741 Number of integer partitions of 2n with reverse-alternating sum -2.

0, 0, 1, 2, 4, 8, 14, 24, 39, 62, 95, 144, 212, 309, 442, 626, 873, 1209, 1653, 2245, 3019, 4035, 5348, 7051, 9229, 12022, 15565, 20063, 25722, 32847, 41746, 52862, 66657, 83768, 104873, 130889, 162797, 201902, 249620, 307789, 378428, 464122, 567721, 692828, 843448

Offset: 0

Author

Gus Wiseman, Jun 08 2021

Keywords

• nonn

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part, so a(n) is the number of integer partitions of 2n with exactly two odd parts, neither of which is the greatest.

Also the number of reversed integer partitions of 2n with alternating sum -2.

Examples

			The a(2) = 1 through a(6) = 14 partitions:
  (31)  (42)    (53)      (64)        (75)
        (3111)  (3221)    (3331)      (4332)
                (4211)    (4222)      (4431)
                (311111)  (4321)      (5322)
                          (5311)      (5421)
                          (322111)    (6411)
                          (421111)    (322221)
                          (31111111)  (333111)
                                      (422211)
                                      (432111)
                                      (531111)
                                      (32211111)
                                      (42111111)
                                      (3111111111)
		

Crossrefs

The version for -1 instead of -2 is A000070.

The non-reversed negative version is A000097.

The ordered version appears to be A001700.

The version for 1 instead of -2 is A035363.

The whole set of partitions of 2n is counted by A058696.

The strict case appears to be A065033.

The version for -1 instead of -2 is A306145.

The version for 2 instead of -2 is A344613.

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

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

A124754 gives alternating sums of standard compositions (reverse: A344618).

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

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

A344611 counts partitions of 2n with reverse-alternating sum >= 0.

Cf. A001250, A003242, A006330, A027187, A028260, A344604, A344607, A344608, A344650, A344651, A344654, A344739.

Programs

Mathematica

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]==-2&]],{n,0,30,2}]
- or -
Table[Length[Select[IntegerPartitions[n],EvenQ[Max[#]]&&Count[#,_?OddQ]==2&]],{n,0,30,2}]

Extensions

More terms from Bert Dobbelaere, Jun 12 2021