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 A358833 Number of rectangular twice-partitions of n of type (P,R,P). 4
 1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1000 Gus Wiseman, Sequences enumerating triangles of integer partitions FORMULA a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022 EXAMPLE The a(1) = 1 through a(5) = 8 twice-partitions: (1) (2) (3) (4) (5) (11) (21) (22) (32) (1)(1) (111) (31) (41) (1)(1)(1) (211) (221) (1111) (311) (2)(2) (2111) (11)(11) (11111) (1)(1)(1)(1) (1)(1)(1)(1)(1) MATHEMATICA twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}]; Table[Length[Select[twiptn[n], SameQ@@Length/@#&&SameQ@@Total/@#&]], {n, 0, 10}] PROG (PARI) P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))} 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 CROSSREFS This is the rectangular case of A279787. This is the case of A306319 with constant sums. For distinct instead of constant lengths and sums we have A358832. The version for multiset partitions of integer partitions is A358835. A063834 counts twice-partitions, strict A296122, row-sums of A321449. A281145 counts same-trees. Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831. Sequence in context: A030014 A047968 A358835 * A322117 A181778 A245026 Adjacent sequences: A358830 A358831 A358832 * A358834 A358835 A358836 KEYWORD nonn AUTHOR Gus Wiseman, Dec 04 2022 EXTENSIONS Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022 STATUS approved

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Last modified April 12 07:56 EDT 2024. Contains 371626 sequences. (Running on oeis4.)