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 A365662 #14 Apr 24 2025 06:23:37
%S A365662 1,0,0,2,2,6,8,14,18,32,42,66,92,136,190,280,374,532,744,1014,1366,
%T A365662 1896,2512,3384,4526,6006,7910,10496,13648,17842,23338,30116,38826,
%U A365662 50256,64298,82258,105156,133480,169392,214778,270620,340554,428772,536302,670522
%N A365662 Number of ordered pairs of disjoint strict integer partitions of n.
%C A365662 Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.
%H A365662 Vaclav Kotesovec, <a href="/A365662/b365662.txt">Table of n, a(n) for n = 0..1000</a>
%F A365662 a(n) = 2*A108796(n) for n > 1.
%F A365662 a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - _Ilya Gutkovskiy_, Apr 24 2025
%e A365662 The a(0) = 1 through a(7) = 14 pairs:
%e A365662 ()() . . (21)(3) (31)(4) (32)(5) (42)(6) (43)(7)
%e A365662 (3)(21) (4)(31) (41)(5) (51)(6) (52)(7)
%e A365662 (5)(32) (6)(42) (61)(7)
%e A365662 (5)(41) (6)(51) (7)(43)
%e A365662 (32)(41) (321)(6) (7)(52)
%e A365662 (41)(32) (42)(51) (7)(61)
%e A365662 (51)(42) (421)(7)
%e A365662 (6)(321) (43)(52)
%e A365662 (43)(61)
%e A365662 (52)(43)
%e A365662 (52)(61)
%e A365662 (61)(43)
%e A365662 (61)(52)
%e A365662 (7)(421)
%t A365662 Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
%t A365662 Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* _Vaclav Kotesovec_, Apr 24 2025 *)
%Y A365662 For subsets instead of partitions we have A000244, non-disjoint A000302.
%Y A365662 If the partitions can have different sums we get A032302.
%Y A365662 The non-strict version is A054440, non-disjoint A001255.
%Y A365662 The unordered version is A108796, non-strict A260669.
%Y A365662 A000041 counts integer partitions, strict A000009.
%Y A365662 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365662 A000712 counts distinct submultisets of partitions.
%Y A365662 A002219 and A237258 count partitions of 2n including a partition of n.
%Y A365662 A304792 counts subset-sums of partitions, positive A276024, strict A284640.
%Y A365662 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365662 Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.
%K A365662 nonn
%O A365662 0,4
%A A365662 _Gus Wiseman_, Sep 19 2023