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%I #4 Mar 26 2018 20:02:19
%S 1,1,2,4,9,19,43,91,201,422,918,1896,4089,8376,17793,36445,76446,
%T 155209,324481,655426,1355220,2741092,5617505,11291037,23086423,
%U 46227338,93753196,187754647,378675055,754695631,1518414812,3016719277,6037006608,11984729983
%N Number of rooted thrice-partitions of n.
%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. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.
%e The a(5) = 9 rooted thrice-partitions:
%e ((2)), ((11)), ((1)()), (()()()),
%e ((1))(), (()())(), (())(()),
%e (())()(),
%e ()()()().
%e The a(6) = 19 rooted thrice-partitions:
%e ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
%e ((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
%e ((1))()(), (()())()(), (())(())(),
%e (())()()(),
%e ()()()()().
%t twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
%t thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
%t Array[thrire,30]
%Y Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 25 2018
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