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

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.

Sequence in context: A011911 A175822 A078346 * A122051 A073471 A308825

Adjacent sequences: A301757 A301758 A301759 * A301761 A301762 A301763

Keyword

nonn

Author

Gus Wiseman, Mar 26 2018

Status

approved