cp's OEIS Frontend

A363532 Number of integer partitions of n with weighted alternating sum 0.

%I A363532 #6 Jun 15 2023 20:06:40
%S A363532 1,0,0,1,0,0,2,2,0,3,3,3,5,5,10,12,7,14,25,18,22,48,48,41,67,82,89,
%T A363532 111,140,170,220,214,264,392,386,436,623,693,756,934,1102,1301,1565,
%U A363532 1697,2132,2616,2727,3192,4062,4550,5000,6132,7197,8067,9338,10750,12683
%N A363532 Number of integer partitions of n with weighted alternating sum 0.
%C A363532 We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) * i * y_i.
%e A363532 The a(11) = 3 through a(15) = 12 partitions (A = 10):
%e A363532   (33221)   (84)        (751)       (662)       (A5)
%e A363532   (44111)   (6222)      (5332)      (4442)      (8322)
%e A363532   (222221)  (7311)      (6421)      (5531)      (9411)
%e A363532             (621111)    (532111)    (43331)     (722211)
%e A363532             (51111111)  (42211111)  (54221)     (831111)
%e A363532                                     (65111)     (3322221)
%e A363532                                     (432221)    (3333111)
%e A363532                                     (443111)    (4422111)
%e A363532                                     (32222111)  (5511111)
%e A363532                                     (33311111)  (22222221)
%e A363532                                                 (72111111)
%e A363532                                                 (6111111111)
%t A363532 altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
%t A363532 Table[Length[Select[IntegerPartitions[n],altwtsum[#]==0&]],{n,0,30}]
%Y A363532 The unweighted version is A035363.
%Y A363532 These partitions have ranks A363621.
%Y A363532 The triangle for this rank statistic is A363623, reverse A363622.
%Y A363532 The version for compositions is A363626.
%Y A363532 A000041 counts integer partitions.
%Y A363532 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363532 A304818 gives weighted sum of prime indices, reverse A318283.
%Y A363532 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A363532 A363619 gives weighted alternating sum of prime indices, reverse A363620.
%Y A363532 A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
%Y A363532 Cf. A008284, A053632, A106529, A261079, A320387, A360672, A360675, A362559.
%K A363532 nonn
%O A363532 0,7
%A A363532 _Gus Wiseman_, Jun 14 2023