A301760 - OEIS

%I #6 Mar 26 2018 20:03:03

%S 1,1,2,4,7,11,17,24,34,46,63,82,109,140,183,233,298,376,479,598,753,

%T 938,1171,1449,1797,2210,2726,3342,4095,4990,6088,7388,8968,10843,

%U 13099,15770,18975,22756,27276,32603,38925,46353,55158,65479,77656,91904,108645

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

%C 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.

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

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

%t nn=50;

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

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 26 2018