cp's OEIS Frontend

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

%I A344741 #10 Jun 12 2021 06:05:02
%S A344741 0,0,1,2,4,8,14,24,39,62,95,144,212,309,442,626,873,1209,1653,2245,
%T A344741 3019,4035,5348,7051,9229,12022,15565,20063,25722,32847,41746,52862,
%U A344741 66657,83768,104873,130889,162797,201902,249620,307789,378428,464122,567721,692828,843448
%N A344741 Number of integer partitions of 2n with reverse-alternating sum -2.
%C A344741 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.
%C A344741 Also the number of reversed integer partitions of 2n with alternating sum -2.
%e A344741 The a(2) = 1 through a(6) = 14 partitions:
%e A344741   (31)  (42)    (53)      (64)        (75)
%e A344741         (3111)  (3221)    (3331)      (4332)
%e A344741                 (4211)    (4222)      (4431)
%e A344741                 (311111)  (4321)      (5322)
%e A344741                           (5311)      (5421)
%e A344741                           (322111)    (6411)
%e A344741                           (421111)    (322221)
%e A344741                           (31111111)  (333111)
%e A344741                                       (422211)
%e A344741                                       (432111)
%e A344741                                       (531111)
%e A344741                                       (32211111)
%e A344741                                       (42111111)
%e A344741                                       (3111111111)
%t A344741 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344741 Table[Length[Select[IntegerPartitions[n],sats[#]==-2&]],{n,0,30,2}]
%t A344741 - or -
%t A344741 Table[Length[Select[IntegerPartitions[n],EvenQ[Max[#]]&&Count[#,_?OddQ]==2&]],{n,0,30,2}]
%Y A344741 The version for -1 instead of -2 is A000070.
%Y A344741 The non-reversed negative version is A000097.
%Y A344741 The ordered version appears to be A001700.
%Y A344741 The version for 1 instead of -2 is A035363.
%Y A344741 The whole set of partitions of 2n is counted by A058696.
%Y A344741 The strict case appears to be A065033.
%Y A344741 The version for -1 instead of -2 is A306145.
%Y A344741 The version for 2 instead of -2 is A344613.
%Y A344741 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344741 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344741 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A344741 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344741 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344741 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A344741 Cf. A001250, A003242, A006330, A027187, A028260, A344604, A344607, A344608, A344650, A344651, A344654, A344739.
%K A344741 nonn
%O A344741 0,4
%A A344741 _Gus Wiseman_, Jun 08 2021
%E A344741 More terms from _Bert Dobbelaere_, Jun 12 2021