This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360675 #12 Mar 11 2023 23:06:22
%S A360675 1,1,0,1,1,0,1,2,0,0,1,2,2,0,0,1,3,3,0,0,0,1,3,5,2,0,0,0,1,4,6,4,0,0,
%T A360675 0,0,1,4,9,5,3,0,0,0,0,1,5,10,10,4,0,0,0,0,0,1,5,13,12,9,2,0,0,0,0,0,
%U A360675 1,6,15,18,11,5,0,0,0,0,0,0
%N A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.
%C A360675 Also the number of integer partitions of n whose left half (inclusive) sums to n-k.
%e A360675 Triangle begins:
%e A360675 1
%e A360675 1 0
%e A360675 1 1 0
%e A360675 1 2 0 0
%e A360675 1 2 2 0 0
%e A360675 1 3 3 0 0 0
%e A360675 1 3 5 2 0 0 0
%e A360675 1 4 6 4 0 0 0 0
%e A360675 1 4 9 5 3 0 0 0 0
%e A360675 1 5 10 10 4 0 0 0 0 0
%e A360675 1 5 13 12 9 2 0 0 0 0 0
%e A360675 1 6 15 18 11 5 0 0 0 0 0 0
%e A360675 1 6 18 22 20 6 4 0 0 0 0 0 0
%e A360675 1 7 20 29 26 13 5 0 0 0 0 0 0 0
%e A360675 1 7 24 34 37 19 11 2 0 0 0 0 0 0 0
%e A360675 1 8 26 44 46 30 16 5 0 0 0 0 0 0 0 0
%e A360675 1 8 30 50 63 40 27 8 4 0 0 0 0 0 0 0 0
%e A360675 1 9 33 61 75 61 36 15 6 0 0 0 0 0 0 0 0 0
%e A360675 1 9 37 70 96 75 61 21 12 3 0 0 0 0 0 0 0 0 0
%e A360675 For example, row n = 9 counts the following partitions:
%e A360675 (9) (81) (72) (63) (54)
%e A360675 (441) (432) (333) (3222)
%e A360675 (531) (522) (3321) (21111111)
%e A360675 (621) (4311) (4221) (111111111)
%e A360675 (711) (5211) (22221)
%e A360675 (6111) (222111)
%e A360675 (32211) (321111)
%e A360675 (33111) (411111)
%e A360675 (42111) (2211111)
%e A360675 (51111) (3111111)
%e A360675 For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).
%t A360675 Table[Length[Select[IntegerPartitions[n], Total[Take[#,-Floor[Length[#]/2]]]==k&]],{n,0,18},{k,0,n}]
%Y A360675 The central diagonal T(2n,n) is A000005.
%Y A360675 Row sums are A000041.
%Y A360675 Diagonal sums are A360671, exclusive A360673.
%Y A360675 The right inclusive version is A360672 with rows reversed.
%Y A360675 The left version has central diagonal A360674, ranks A360953.
%Y A360675 A008284 counts partitions by length.
%Y A360675 A359893 and A359901 count partitions by median.
%Y A360675 First for prime indices, second for partitions, third for prime factors:
%Y A360675 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360675 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360675 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360675 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360675 Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616.
%K A360675 nonn,tabl
%O A360675 0,8
%A A360675 _Gus Wiseman_, Feb 27 2023