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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
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.
Sequence in context: A030014 A047968 A358835 * A322117 A181778 A245026
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.)