cp's OEIS Frontend

A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

%I A344610 #10 Jun 09 2021 06:22:53
%S A344610 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 A344610 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 A344610 62,38,21,12,6,3,1,1,56,109,140,103,63,38,21,12,6,3,1,1
%N A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.
%C A344610 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 A344610 Also the number of reversed integer partitions of 2n with alternating sum 2k.
%e A344610 Triangle begins:
%e A344610    1
%e A344610    1   1
%e A344610    2   1   1
%e A344610    3   3   1   1
%e A344610    5   5   3   1   1
%e A344610    7   9   6   3   1   1
%e A344610   11  14  12   6   3   1   1
%e A344610   15  23  20  12   6   3   1   1
%e A344610   22  34  35  21  12   6   3   1   1
%e A344610   30  52  56  38  21  12   6   3   1   1
%e A344610   42  75  91  62  38  21  12   6   3   1   1
%e A344610   56 109 140 103  63  38  21  12   6   3   1   1
%e A344610   77 153 215 163 106  63  38  21  12   6   3   1   1
%e A344610 Row n = 5 counts the following partitions:
%e A344610   (55)          (442)        (433)      (622)    (811)  (10)
%e A344610   (3322)        (541)        (532)      (721)
%e A344610   (4411)        (22222)      (631)      (61111)
%e A344610   (222211)      (32221)      (42211)
%e A344610   (331111)      (33211)      (52111)
%e A344610   (22111111)    (43111)      (4111111)
%e A344610   (1111111111)  (2221111)
%e A344610                 (3211111)
%e A344610                 (211111111)
%t A344610 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344610 Table[Length[Select[IntegerPartitions[n],k==sats[#]&]],{n,0,15,2},{k,0,n,2}]
%Y A344610 The columns with initial 0's removed appear to converge to A006330.
%Y A344610 The odd version is A239829.
%Y A344610 The non-reversed version is A239830.
%Y A344610 Row sums are A344611, odd bisection of A344607.
%Y A344610 Including odd n and negative k gives A344612 (strict: A344739).
%Y A344610 The strict case is A344649 (row sums: A344650).
%Y A344610 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344610 A103919 counts partitions by sum and alternating sum.
%Y A344610 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A344610 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344610 A325534/A325535 count separable/inseparable partitions.
%Y A344610 A344604 counts wiggly compositions with twins.
%Y A344610 A344618 gives reverse-alternating sums of standard compositions.
%Y A344610 Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.
%K A344610 nonn,tabl
%O A344610 0,4
%A A344610 _Gus Wiseman_, May 31 2021