%I #10 Dec 31 2022 14:52:23

%S 1,1,3,4,8,8,17,16,32,34,56,57,119,102,179,199,335,298,598,491,960,

%T 925,1441,1256,2966,2026,3726,3800,6488,4566,11726,6843,16176,14109,

%U 21824,16688,49507,21638,50286,50394,99408,44584,165129,63262,208853,205109,248150

%N Number of rectangular twice-partitions of n of type (P,R,P).

%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n, so these are twice-partitions of n into partitions with constant lengths and constant sums.

%H Andrew Howroyd, <a href="/A358833/b358833.txt">Table of n, a(n) for n = 0..1000</a>

%H Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>

%F a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - _Andrew Howroyd_, Dec 31 2022

%e The a(1) = 1 through a(5) = 8 twice-partitions:

%e (1) (2) (3) (4) (5)

%e (11) (21) (22) (32)

%e (1)(1) (111) (31) (41)

%e (1)(1)(1) (211) (221)

%e (1111) (311)

%e (2)(2) (2111)

%e (11)(11) (11111)

%e (1)(1)(1)(1) (1)(1)(1)(1)(1)

%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];

%t Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{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(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ _Andrew Howroyd_, Dec 31 2022

%Y This is the rectangular case of A279787.

%Y This is the case of A306319 with constant sums.

%Y For distinct instead of constant lengths and sums we have A358832.

%Y The version for multiset partitions of integer partitions is A358835.

%Y A063834 counts twice-partitions, strict A296122, row-sums of A321449.

%Y A281145 counts same-trees.

%Y Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 04 2022

%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2022