%I #10 Jun 09 2021 06:22:53
%S 1,1,1,2,1,1,3,3,1,1,5,5,3,1,1,7,9,6,3,1,1,11,14,12,6,3,1,1,15,23,20,
%T 12,6,3,1,1,22,34,35,21,12,6,3,1,1,30,52,56,38,21,12,6,3,1,1,42,75,91,
%U 62,38,21,12,6,3,1,1,56,109,140,103,63,38,21,12,6,3,1,1
%N Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.
%C The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.
%C Also the number of reversed integer partitions of 2n with alternating sum 2k.
%e Triangle begins:
%e 1
%e 1 1
%e 2 1 1
%e 3 3 1 1
%e 5 5 3 1 1
%e 7 9 6 3 1 1
%e 11 14 12 6 3 1 1
%e 15 23 20 12 6 3 1 1
%e 22 34 35 21 12 6 3 1 1
%e 30 52 56 38 21 12 6 3 1 1
%e 42 75 91 62 38 21 12 6 3 1 1
%e 56 109 140 103 63 38 21 12 6 3 1 1
%e 77 153 215 163 106 63 38 21 12 6 3 1 1
%e Row n = 5 counts the following partitions:
%e (55) (442) (433) (622) (811) (10)
%e (3322) (541) (532) (721)
%e (4411) (22222) (631) (61111)
%e (222211) (32221) (42211)
%e (331111) (33211) (52111)
%e (22111111) (43111) (4111111)
%e (1111111111) (2221111)
%e (3211111)
%e (211111111)
%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t Table[Length[Select[IntegerPartitions[n],k==sats[#]&]],{n,0,15,2},{k,0,n,2}]
%Y The columns with initial 0's removed appear to converge to A006330.
%Y The odd version is A239829.
%Y The non-reversed version is A239830.
%Y Row sums are A344611, odd bisection of A344607.
%Y Including odd n and negative k gives A344612 (strict: A344739).
%Y The strict case is A344649 (row sums: A344650).
%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A103919 counts partitions by sum and alternating sum.
%Y A120452 counts partitions of 2n with rev-alt 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 A344618 gives reverse-alternating sums of standard compositions.
%Y Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.
%K nonn,tabl
%O 0,4
%A _Gus Wiseman_, May 31 2021