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 A355389 #13 Feb 07 2024 21:07:47
%S A355389 0,0,1,3,10,21,55,105,231,435,861,1540,2926,5050,9045,15400,26565,
%T A355389 43956,73920,119805,196251,313236,501501,786885,1239525,1915903,
%U A355389 2965830,4528545,6909903,10417330,15699606,23403061,34848726,51435153,75761895,110744403,161577276
%N A355389 Number of unordered pairs of distinct integer partitions of n.
%F A355389 a(n) = binomial(A000041(n), 2) = A355390(n)/2.
%e A355389 The a(0) = 0 through a(4) = 10 pairs:
%e A355389 . . (2)(11) (3)(21) (4)(22)
%e A355389 (3)(111) (4)(31)
%e A355389 (21)(111) (22)(31)
%e A355389 (4)(211)
%e A355389 (22)(211)
%e A355389 (31)(211)
%e A355389 (4)(1111)
%e A355389 (22)(1111)
%e A355389 (31)(1111)
%e A355389 (211)(1111)
%p A355389 a:= n-> binomial(combinat[numbpart](n),2):
%p A355389 seq(a(n), n=0..36); # _Alois P. Heinz_, Feb 07 2024
%t A355389 Table[Binomial[PartitionsP[n],2],{n,0,6}]
%o A355389 (PARI) a(n) = binomial(numbpart(n), 2); \\ _Michel Marcus_, Jul 05 2022
%Y A355389 The version for compositions is A006516.
%Y A355389 Without distinctness we get A086737.
%Y A355389 The unordered version is A355390, without distinctness A001255.
%Y A355389 A000041 counts partitions, strict A000009.
%Y A355389 A001970 counts multiset partitions of partitions.
%Y A355389 A063834 counts partitions of each part of a partition.
%Y A355389 Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.
%K A355389 nonn
%O A355389 0,4
%A A355389 _Gus Wiseman_, Jul 04 2022