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A301480
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Number of rooted twice-partitions of n.
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19
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1, 1, 2, 4, 8, 15, 30, 54, 103, 186, 345, 606, 1115, 1936, 3466, 6046, 10630, 18257, 31927, 54393, 93894, 159631, 272155, 458891, 779375, 1305801, 2196009, 3667242, 6130066, 10170745, 16923127, 27942148, 46211977, 76039205, 125094369, 204952168, 335924597
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OFFSET
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1,3
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COMMENTS
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A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.
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LINKS
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FORMULA
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O.g.f.: x * Product_{n > 0} 1/(1 - P(n-1) x^n) where P = A000041.
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EXAMPLE
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The a(5) = 8 rooted twice-partitions: ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()).
The a(6) = 15 rooted twice-partitions:
(4), (31), (22), (211), (1111),
(3)(), (21)(), (111)(), (2)(1), (11)(1),
(2)()(), (11)()(), (1)(1)(),
(1)()()(),
()()()()().
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MATHEMATICA
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nn=30;
ser=x*Product[1/(1-PartitionsP[n-1]x^n), {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]
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PROG
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(PARI) seq(n)={Vec(1/prod(k=1, n-1, 1 - numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018
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CROSSREFS
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Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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