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

Author

Gus Wiseman, Mar 26 2018

Keywords

• nonn

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.

Examples

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

Crossrefs

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

Programs

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]

Formula

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