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%I #6 Mar 26 2018 20:02:27
%S 1,1,2,3,5,8,12,18,29,42,61,86,127,181,257,352,489,668,935,1270,1730,
%T 2312,3101,4112,5533,7345,9742,12785,16793,21821,28452,36908,48108,
%U 62198,80337,103081,132372,168805,215247,273678
%N Number of rooted twice-partitions of n where the composite rooted partition is strict.
%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(8) = 18 rooted twice-partitions where the composite rooted partition is strict:
%e (6), (51), (42), (321),
%e (5)(), (41)(), (32)(), (4)(1), (3)(2),
%e (4)()(), (31)()(), (3)(1)(),
%e (3)()()(), (21)()()(), (2)(1)()(),
%e (2)()()()(),
%e (1)()()()()(),
%e ()()()()()()().
%t twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
%t Table[Select[twirtns[n],UnsameQ@@Join@@#&]//Length,{n,30}]
%Y Cf. A002865, A063834, A093637, A127524, A279790, A294788, A301422, A301462, A301467, A301480, A301706.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 26 2018
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