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A358833 Number of rectangular twice-partitions of n of type (P,R,P).

%I A358833 #10 Dec 31 2022 14:52:23
%S A358833 1,1,3,4,8,8,17,16,32,34,56,57,119,102,179,199,335,298,598,491,960,
%T A358833 925,1441,1256,2966,2026,3726,3800,6488,4566,11726,6843,16176,14109,
%U A358833 21824,16688,49507,21638,50286,50394,99408,44584,165129,63262,208853,205109,248150
%N A358833 Number of rectangular twice-partitions of n of type (P,R,P).
%C A358833 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 A358833 Andrew Howroyd, <a href="/A358833/b358833.txt">Table of n, a(n) for n = 0..1000</a>
%H A358833 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F A358833 a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n  > 0. - _Andrew Howroyd_, Dec 31 2022
%e A358833 The a(1) = 1 through a(5) = 8 twice-partitions:
%e A358833   (1)  (2)     (3)        (4)           (5)
%e A358833        (11)    (21)       (22)          (32)
%e A358833        (1)(1)  (111)      (31)          (41)
%e A358833                (1)(1)(1)  (211)         (221)
%e A358833                           (1111)        (311)
%e A358833                           (2)(2)        (2111)
%e A358833                           (11)(11)      (11111)
%e A358833                           (1)(1)(1)(1)  (1)(1)(1)(1)(1)
%t A358833 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358833 Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]
%o A358833 (PARI)
%o A358833 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358833 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 A358833 This is the rectangular case of A279787.
%Y A358833 This is the case of A306319 with constant sums.
%Y A358833 For distinct instead of constant lengths and sums we have A358832.
%Y A358833 The version for multiset partitions of integer partitions is A358835.
%Y A358833 A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A358833 A281145 counts same-trees.
%Y A358833 Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.
%K A358833 nonn
%O A358833 0,3
%A A358833 _Gus Wiseman_, Dec 04 2022
%E A358833 Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2022