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 A362051 #17 Apr 28 2023 15:00:22
%S A362051 1,1,2,6,11,27,44,93,149,271,432,744,1109,1849,2764,4287,6328,9673,
%T A362051 13853,20717,29343,42609,60100,85893,118475,167453,230080,318654,
%U A362051 433763,595921,800878,1090189,1456095,1957032,2600199,3465459,4558785,6041381,7908681
%N A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.
%C A362051 Even bisection of A362558.
%C A362051 a(0) = 1; a(n) = A000041(2n) - A322439(n). - _Alois P. Heinz_, Apr 27 2023
%e A362051 The a(1) = 1 through a(4) = 11 partitions:
%e A362051 (2) (4) (6) (8)
%e A362051 (31) (42) (53)
%e A362051 (51) (62)
%e A362051 (222) (71)
%e A362051 (411) (332)
%e A362051 (2211) (521)
%e A362051 (611)
%e A362051 (3221)
%e A362051 (3311)
%e A362051 (5111)
%e A362051 (32111)
%e A362051 The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).
%t A362051 Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]
%Y A362051 The version for compositions is A000302, bisection of A213173.
%Y A362051 The complement is counted by A322439.
%Y A362051 Even bisection of A362558.
%Y A362051 A000041 counts integer partitions, strict A000009.
%Y A362051 A304442 counts partitions with all equal run-sums.
%Y A362051 A325347 counts partitions with integer median, complement A307683.
%Y A362051 A353836 counts partitions by number of distinct run-sums.
%Y A362051 A359893/A359901/A359902 count partitions by median.
%Y A362051 Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.
%K A362051 nonn
%O A362051 0,3
%A A362051 _Gus Wiseman_, Apr 24 2023