cp's OEIS Frontend

A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

%I A358194 #6 Jan 01 2023 19:30:53
%S A358194 1,1,1,1,1,0,1,1,1,0,1,1,0,1,1,1,0,0,1,1,0,1,1,0,1,1,1,0,0,1,1,1,1,0,
%T A358194 1,1,0,1,1,0,1,1,1,0,0,0,1,1,1,0,1,1,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,
%U A358194 1,1,1,1,0,1,1,1,2,1,0,1,1,1,1,1,0,1,1,0,1,1,0,1,1
%N A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.
%C A358194 The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).
%e A358194 Triangle begins:
%e A358194   1
%e A358194   1
%e A358194   1 1
%e A358194   1 0 1 1
%e A358194   1 0 1 1 0 1 1
%e A358194   1 0 0 1 1 0 1 1 0 1 1
%e A358194   1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
%e A358194   1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
%e A358194   1 0 0 0 1 1 1 1 0 1 1 1 2 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
%e A358194 For example, the T(15,59) = 5 partitions are: (8,2,2,2,1), (7,3,3,1,1), (6,5,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1).
%t A358194 Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==k&]],{n,0,8},{k,n,n*(n+1)/2}]
%Y A358194 Row sums are A000041.
%Y A358194 The version for compositions is A053632.
%Y A358194 Row lengths are A152947.
%Y A358194 The version for reversed partitions is A264034.
%Y A358194 A048793 = partial sums of reversed standard compositions, sum A029931.
%Y A358194 A358134 = partial sums of standard compositions, sum A359042.
%Y A358194 A358136 = partial sums of prime indices, sum A318283.
%Y A358194 A359361 = partial sums of reversed prime indices, sum A304818.
%Y A358194 Cf. A000009, A325362, A358137, A359397.
%K A358194 nonn,tabf
%O A358194 0,77
%A A358194 _Gus Wiseman_, Dec 31 2022