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Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).
2

%I #5 Mar 26 2018 20:03:39

%S 1,1,2,2,3,3,4,5,8,7,11,11,19,16,27,23,42,33,63,47,87,71,119,90,195

%N Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).

%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.

%e The a(9) = 8 rooted twice-partitions:

%e (7), (61), (52), (43), (421),

%e (3)(21), (21)(3),

%e ()()()()()()()().

%t twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];

%t Table[Select[twirtns[n],SameQ@@Total/@#&&UnsameQ@@Join@@#&]//Length,{n,20}]

%Y Cf. A002865, A032305, A063834, A093637, A127524, A279791, A296133, A301422, A301462, A301467, A301480, A301706.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Mar 26 2018