A301480 Number of rooted twice-partitions of n.

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

Offset

1,3

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

O.g.f.: x * Product_{n > 0} 1/(1 - P(n-1) x^n) where P = A000041.

Example

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)()()(),
()()()()().

Mathematica

nn=30;
ser=x*Product[1/(1-PartitionsP[n-1]x^n), {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]

Prog

(PARI) seq(n)={Vec(1/prod(k=1, n-1, 1 - numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Crossrefs

Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.

Sequence in context: A262333 A293335 A344687 * A217777 A034338 A166861

Adjacent sequences: A301477 A301478 A301479 * A301481 A301482 A301483

Keyword

nonn

Author

Gus Wiseman, Mar 22 2018

Status

approved