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 A336102 #20 Apr 08 2021 07:24:54
%S A336102 0,0,1,1,3,3,8,8,20,20,48,48,112,112,256,256,576,576,1280,1280,2816,
%T A336102 2816,6144,6144,13312,13312,28672,28672,61440,61440,131072,131072,
%U A336102 278528,278528,589824,589824,1245184,1245184,2621440,2621440,5505024,5505024,11534336
%N A336102 Number of inseparable multisets of size n covering an initial interval of positive integers.
%C A336102 A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
%C A336102 Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one.
%C A336102 Also the number of compositions of n whose greatest part is greater than the sum of its remaining parts plus one. For example, the a(2) = 1 through a(7) = 8 compositions are:
%C A336102 (2) (3) (4) (5) (6) (7)
%C A336102 (1,3) (1,4) (1,5) (1,6)
%C A336102 (3,1) (4,1) (2,4) (2,5)
%C A336102 (4,2) (5,2)
%C A336102 (5,1) (6,1)
%C A336102 (1,1,4) (1,1,5)
%C A336102 (1,4,1) (1,5,1)
%C A336102 (4,1,1) (5,1,1)
%H A336102 Michael De Vlieger, <a href="/A336102/b336102.txt">Table of n, a(n) for n = 0..6625</a>
%H A336102 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-4).
%F A336102 a(2*n) = a(2*n + 1) = A049610(n + 1).
%F A336102 a(n) = 2^(n-1) - A336103(n).
%F A336102 A001792 repeated for n > 1. _David A. Corneth_, Jul 09 2020
%F A336102 From _Chai Wah Wu_, Apr 07 2021: (Start)
%F A336102 a(n) = 4*a(n-2) - 4*a(n-4) for n > 5.
%F A336102 G.f.: x^2*(1 - x)*(x + 1)^2/(2*x^2 - 1)^2. (End)
%e A336102 The a(2) = 1 through a(7) = 8 multisets:
%e A336102 {11} {111} {1111} {11111} {111111} {1111111}
%e A336102 {1112} {11112} {111112} {1111112}
%e A336102 {1222} {12222} {111122} {1111122}
%e A336102 {111123} {1111123}
%e A336102 {112222} {1122222}
%e A336102 {122222} {1222222}
%e A336102 {122223} {1222223}
%e A336102 {123333} {1233333}
%t A336102 Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],With[{mx=Max@@#},mx>1+Total[DeleteCases[#,mx,{1},1]]]&]],{n,0,15}]
%t A336102 (* Second program: *)
%t A336102 CoefficientList[Series[x^2*(1 - x) (x + 1)^2/(2 x^2 - 1)^2, {x, 0, 43}], x] (* _Michael De Vlieger_, Apr 07 2021 *)
%Y A336102 The strong (weakly decreasing multiplicities) case is A025065.
%Y A336102 The bisection is A049610.
%Y A336102 The separable version is A336103.
%Y A336102 Sequences covering an initial interval are A000670.
%Y A336102 Anti-run compositions are A003242.
%Y A336102 Anti-run patterns are A005649.
%Y A336102 Separable partitions are A325534.
%Y A336102 Inseparable partitions are A325535.
%Y A336102 Inseparable factorizations are A333487.
%Y A336102 Anti-run compositions are ranked by A333489.
%Y A336102 Heinz numbers of inseparable partitions are A335448.
%Y A336102 Cf. A001792, A019472, A052841, A106351, A124767, A269134, A292884, A335433, A335126, A335452, A335548.
%K A336102 nonn
%O A336102 0,5
%A A336102 _Gus Wiseman_, Jul 08 2020