cp's OEIS Frontend

A363622 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with weighted alternating sum k (leading and trailing 0's omitted).

%I A363622 #7 Jun 15 2023 20:06:52
%S A363622 1,1,1,0,0,1,1,0,1,1,2,0,0,1,0,1,1,2,0,1,1,1,1,1,3,0,0,2,0,1,1,2,1,1,
%T A363622 3,0,2,2,1,1,2,2,1,1,5,0,0,3,0,2,2,2,1,3,2,1,1,5,0,3,3,2,2,3,2,2,4,2,
%U A363622 1,1,7,0,0,5,0,3,3,4,2,4,2,4,4,2,1,1
%N A363622 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with weighted alternating sum k (leading and trailing 0's omitted).
%C A363622 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. For example:
%C A363622 - (3,3,2,1,1) has weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 = 4.
%C A363622 - (1,2,2,3) has weighted alternating sum  1*1 - 2*2 + 3*2 - 4*3 = -9.
%e A363622 Triangle begins:
%e A363622   1
%e A363622   1
%e A363622   1  0  0  1
%e A363622   1  0  1  1
%e A363622   2  0  0  1  0  1  1
%e A363622   2  0  1  1  1  1  1
%e A363622   3  0  0  2  0  1  1  2  1  1
%e A363622   3  0  2  2  1  1  2  2  1  1
%e A363622   5  0  0  3  0  2  2  2  1  3  2  1  1
%e A363622   5  0  3  3  2  2  3  2  2  4  2  1  1
%e A363622   7  0  0  5  0  3  3  4  2  4  2  4  4  2  1  1
%e A363622   7  0  5  5  3  3  5  4  3  5  3  5  4  2  1  1
%e A363622 Row n = 6 counts the following partitions:
%e A363622   k=-3            k=0        k=2    k=3   k=4      k=5    k=6
%e A363622   -----------------------------------------------------------
%e A363622   (33)      .  .  (42)    .  (321)  (51)  (222)    (411)  (6)
%e A363622   (2211)          (3111)                  (21111)
%e A363622   (111111)
%t A363622 altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
%t A363622 Table[Length[Select[IntegerPartitions[n],altwtsum[#]==k&]],{n,0,15},{k,Min[altwtsum/@IntegerPartitions[n]], Max[altwtsum/@IntegerPartitions[n]]}]
%Y A363622 Row sums are A000041.
%Y A363622 The unweighted version is A103919 with leading zeros removed.
%Y A363622 Row-lengths appear to be A168233.
%Y A363622 Central column T(n,0) is A363532, ranks A363621.
%Y A363622 The corresponding rank statistic is A363619, reverse A363620.
%Y A363622 The reverse version is A363623.
%Y A363622 A053632 counts compositions by weighted sum.
%Y A363622 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363622 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A363622 A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
%Y A363622 Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363626.
%K A363622 nonn,tabf
%O A363622 0,11
%A A363622 _Gus Wiseman_, Jun 15 2023