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 A360672 #13 Mar 11 2023 23:06:27
%S A360672 1,1,0,1,1,0,1,1,1,0,1,0,3,1,0,1,0,2,3,1,0,1,0,1,4,4,1,0,1,0,0,3,6,4,
%T A360672 1,0,1,0,0,1,7,7,5,1,0,1,0,0,1,4,8,10,5,1,0,1,0,0,0,3,6,14,11,6,1,0,1,
%U A360672 0,0,0,1,5,12,16,14,6,1,0
%N A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.
%C A360672 Also the number of integer partitions of n whose right half (inclusive) sums to n-k.
%e A360672 Triangle begins:
%e A360672 1
%e A360672 1 0
%e A360672 1 1 0
%e A360672 1 1 1 0
%e A360672 1 0 3 1 0
%e A360672 1 0 2 3 1 0
%e A360672 1 0 1 4 4 1 0
%e A360672 1 0 0 3 6 4 1 0
%e A360672 1 0 0 1 7 7 5 1 0
%e A360672 1 0 0 1 4 8 10 5 1 0
%e A360672 1 0 0 0 3 6 14 11 6 1 0
%e A360672 1 0 0 0 1 5 12 16 14 6 1 0
%e A360672 1 0 0 0 1 2 12 14 23 16 7 1 0
%e A360672 1 0 0 0 0 2 7 13 24 27 19 7 1 0
%e A360672 1 0 0 0 0 1 5 9 24 30 35 21 8 1 0
%e A360672 1 0 0 0 0 1 3 7 17 31 42 40 25 8 1 0
%e A360672 1 0 0 0 0 0 2 4 16 23 46 51 51 27 9 1 0
%e A360672 1 0 0 0 0 0 1 3 10 21 37 57 69 57 31 9 1 0
%e A360672 1 0 0 0 0 0 1 2 7 15 34 47 83 81 69 34 10 1 0
%e A360672 For example, row n = 9 counts the following partitions:
%e A360672 (9) . . (333) (432) (54) (63) (72) (81)
%e A360672 (441) (522) (621) (711)
%e A360672 (22221) (531) (3321) (4311)
%e A360672 (111111111) (3222) (4221) (5211)
%e A360672 (32211) (33111) (6111)
%e A360672 (2211111) (42111)
%e A360672 (3111111) (51111)
%e A360672 (21111111) (222111)
%e A360672 (321111)
%e A360672 (411111)
%e A360672 For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).
%t A360672 Table[Length[Select[IntegerPartitions[n], Total[Take[#,Floor[Length[#]/2]]]==k&]],{n,0,10},{k,0,n}]
%Y A360672 Row sums are A000041.
%Y A360672 Column sums are A360673, inclusive A360671.
%Y A360672 The central diagonal T(2n,n) is A360674, ranks A360953.
%Y A360672 The left inclusive version is A360675 with rows reversed.
%Y A360672 A008284 counts partitions by length.
%Y A360672 A359893 and A359901 count partitions by median.
%Y A360672 First for prime indices, second for partitions, third for prime factors:
%Y A360672 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360672 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360672 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360672 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360672 Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616, A360682, A360686.
%K A360672 nonn,tabl
%O A360672 0,13
%A A360672 _Gus Wiseman_, Feb 27 2023