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 A386638 #7 Aug 18 2025 18:48:00
%S A386638 0,0,1,1,2,2,4,4,7,7,12,12,19,19,30,30,45,45,67,67,97,97,139,139,195,
%T A386638 195,272,272,373,373,508,508,684,684,915,915,1212,1212,1597,1597,2087,
%U A386638 2087,2714,2714,3506,3506,4508,4508,5763,5763,7338,7338,9296,9296
%N A386638 Number of integer partitions of n of inseparable type.
%C A386638 A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.
%C A386638 Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.
%F A386638 For n>1, a(n) = A025065(n-2).
%F A386638 a(n) = A000041(n) - A336106(n).
%e A386638 The a(2) = 1 through a(10) = 12 partitions (A=10):
%e A386638 (2) (3) (4) (5) (6) (7) (8) (9) (A)
%e A386638 (31) (41) (42) (52) (53) (63) (64)
%e A386638 (51) (61) (62) (72) (73)
%e A386638 (411) (511) (71) (81) (82)
%e A386638 (521) (621) (91)
%e A386638 (611) (711) (622)
%e A386638 (5111) (6111) (631)
%e A386638 (721)
%e A386638 (811)
%e A386638 (6211)
%e A386638 (7111)
%e A386638 (61111)
%t A386638 Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]
%Y A386638 Reduplication of A000070 shifted right.
%Y A386638 Same as A025065 shifted right twice.
%Y A386638 The Heinz numbers of these partitions are A335126.
%Y A386638 Row sums of A386586.
%Y A386638 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386638 A239455 counts Look-and-Say partitions, inseparable case A386632.
%Y A386638 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386638 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386638 A335434 counts separable factorizations, inseparable A333487.
%Y A386638 A336103 counts normal separable multisets, inseparable A336102.
%Y A386638 A336106 counts separable type partitions, ranks A335127, sums of A386585.
%Y A386638 A386633 counts separable type set partitions, row sums of A386635.
%Y A386638 A386634 counts inseparable type set partitions, row sums of A386636.
%Y A386638 Cf. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.
%K A386638 nonn,new
%O A386638 0,5
%A A386638 _Gus Wiseman_, Aug 14 2025