A301760 - OEIS

A301760

Number of rooted twice-partitions of n where the composite rooted partition is constant.

1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 63, 82, 109, 140, 183, 233, 298, 376, 479, 598, 753, 938, 1171, 1449, 1797, 2210, 2726, 3342, 4095, 4990, 6088, 7388, 8968, 10843, 13099, 15770, 18975, 22756, 27276, 32603, 38925, 46353, 55158, 65479, 77656, 91904, 108645

(list; graph; refs; listen; history; text; internal format)

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

Table of n, a(n) for n=1..47.

FORMULA

O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).

EXAMPLE

The a(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

MATHEMATICA

nn=50;

ser=(1-nn)/(1-x)+Sum[Product[1/(1-x^(d k+1)), {k, 0, nn}], {d, nn}];

CoefficientList[Series[ser, {x, 0, nn}], x]

CROSSREFS

Cf. A002865, A047968, A063834, A093637, A295924, A301422, A301462, A301467, A301480, A301706.