%I #10 Dec 31 2022 11:20:07
%S 1,1,3,6,13,24,49,91,179,341,664,1280,2503,4872,9557,18750,36927,
%T 72800,143880,284660,564093,1118911,2221834,4415417,8781591,17476099,
%U 34799199,69327512,138176461,275503854,549502119,1096327380,2187894634,4367310138,8719509111
%N Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.
%H Andrew Howroyd, <a href="/A358905/b358905.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 + Sum_{k>=1} (1/(1 - [y^k]P(x,y)) - 1) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - _Andrew Howroyd_, Dec 31 2022
%e The a(0) = 1 through a(4) = 13 sequences:
%e () ((1)) ((2)) ((3)) ((4))
%e ((11)) ((21)) ((22))
%e ((1)(1)) ((111)) ((31))
%e ((1)(2)) ((211))
%e ((2)(1)) ((1111))
%e ((1)(1)(1)) ((1)(3))
%e ((2)(2))
%e ((3)(1))
%e ((11)(11))
%e ((1)(1)(2))
%e ((1)(2)(1))
%e ((2)(1)(1))
%e ((1)(1)(1)(1))
%t ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
%t Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]
%o (PARI)
%o P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o seq(n) = {my(g=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ _Andrew Howroyd_, Dec 31 2022
%Y The case of set partitions is A038041.
%Y The version for weakly decreasing lengths is A141199, strictly A358836.
%Y For equal sums instead of lengths we have A279787.
%Y The case of twice-partitions is A306319, distinct A358830.
%Y The unordered version is A319066.
%Y The case of plane partitions is A323429.
%Y The case of constant sums also is A358833.
%Y A055887 counts sequences of partitions with total sum n.
%Y A281145 counts same-trees.
%Y A319169 counts partitions with constant Omega, ranked by A320324.
%Y A358911 counts compositions with constant Omega, distinct A358912.
%Y Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 07 2022
%E Terms a(16) and beyond from _Andrew Howroyd_, Dec 31 2022