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%I #9 Aug 27 2018 01:52:49
%S 1,1,1,3,4,6,7,9,11,13,16,19,22,26,32,36,42,52,59,66,79,93,108,125,
%T 141,162,192,222,248,285,331,375,430,492,555,632,719,816,929,1051,
%U 1177,1327,1510,1701,1908,2146,2408,2705,3035,3388,3792,4257,4751,5284,5894
%N Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).
%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.
%H Andrew Howroyd, <a href="/A301766/b301766.txt">Table of n, a(n) for n = 1..1000</a>
%e The a(9) = 11 rooted twice-partitions:
%e (7), (1111111),
%e (6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
%e (1111)(1)(), (111)(11)().
%t twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
%t Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]
%o (PARI) a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ _Andrew Howroyd_, Aug 26 2018
%Y Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.
%K nonn
%O 1,4
%A _Gus Wiseman_, Mar 26 2018
%E Terms a(26) and beyond from _Andrew Howroyd_, Aug 26 2018
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