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A358905
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Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.
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4
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1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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The a(0) = 1 through a(4) = 13 sequences:
() ((1)) ((2)) ((3)) ((4))
((11)) ((21)) ((22))
((1)(1)) ((111)) ((31))
((1)(2)) ((211))
((2)(1)) ((1111))
((1)(1)(1)) ((1)(3))
((2)(2))
((3)(1))
((11)(11))
((1)(1)(2))
((1)(2)(1))
((2)(1)(1))
((1)(1)(1)(1))
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MATHEMATICA
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ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp], {comp, Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n], SameQ@@Length/@#&]], {n, 0, 10}]
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PROG
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(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
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
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CROSSREFS
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The case of set partitions is A038041.
The version for weakly decreasing lengths is A141199, strictly A358836.
For equal sums instead of lengths we have A279787.
The case of plane partitions is A323429.
The case of constant sums also is A358833.
A055887 counts sequences of partitions with total sum n.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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