cp's OEIS Frontend

A363626 Number of integer compositions of n with weighted alternating sum 0.

%I A363626 #16 Sep 05 2023 15:21:20
%S A363626 1,0,0,1,1,0,2,5,7,8,14,38,64,87,174,373,649,1069,2051,4091,7453,
%T A363626 13276,25260,48990,91378,168890,321661,618323,1169126,2203649,4211163,
%U A363626 8085240,15421171,29390131,56382040,108443047,208077560,399310778
%N A363626 Number of integer compositions of n with weighted alternating sum 0.
%C A363626 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.
%H A363626 Alois P. Heinz, <a href="/A363626/b363626.txt">Table of n, a(n) for n = 0..150</a> (first 51 terms from Max Alekseyev)
%e A363626 The a(3) = 1 through a(10) = 14 compositions:
%e A363626   (21)  (121)  .  (42)    (331)     (242)      (63)       (541)
%e A363626                   (3111)  (1132)    (1331)     (153)      (2143)
%e A363626                           (2221)    (11132)    (4122)     (3232)
%e A363626                           (21121)   (12221)    (5211)     (4321)
%e A363626                           (112111)  (23111)    (13122)    (15112)
%e A363626                                     (121121)   (14211)    (31231)
%e A363626                                     (1112111)  (411111)   (42121)
%e A363626                                                (1311111)  (114112)
%e A363626                                                           (212122)
%e A363626                                                           (213211)
%e A363626                                                           (311221)
%e A363626                                                           (322111)
%e A363626                                                           (3111121)
%e A363626                                                           (21211111)
%t A363626 altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
%t A363626 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],altwtsum[#]==0&]],{n,0,10}]
%Y A363626 The unweighted version is A138364, ranks A344619.
%Y A363626 The version for partitions is A363532, ranks A363621.
%Y A363626 A000041 counts integer partitions.
%Y A363626 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363626 A304818 gives weighted sum of prime indices, reverse A318283.
%Y A363626 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A363626 A363619 gives weighted alternating sum of prime indices, reverse A363620.
%Y A363626 A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
%Y A363626 Cf. A008284, A053632, A106529, A261079, A320387, A360672, A360675, A362559, A363622, A363623.
%K A363626 nonn
%O A363626 0,7
%A A363626 _Gus Wiseman_, Jun 16 2023
%E A363626 Terms a(22) onward from _Max Alekseyev_, Sep 05 2023