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%I #5 Mar 26 2018 20:03:17
%S 1,1,2,4,7,12,21,34,55,90,143,220,347,528,805,1226,1831,2719,4048,
%T 5940,8710,12714,18403,26529,38220,54679,77899,110810,156848,221181,
%U 311635,436705,610597,852125,1184928,1644136,2276551,3142523,4328960,5953523,8167209
%N Number of ways to choose a constant rooted partition of each part in a rooted partition of n.
%C A rooted partition of n is an integer partition of n - 1.
%F O.g.f.: Product_{n>0} 1/(1 - d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.
%e The a(5) = 7 rooted twice-partitions where the latter rooted partitions are constant: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().
%t Table[Sum[Product[If[k===1,1,DivisorSigma[0,k-1]],{k,ptn}],{ptn,IntegerPartitions[n-1]}],{n,20}]
%Y Cf. A002865, A063834, A093637, A279784, A295935, A300383, A301422, A301462, A301467, A301480, A301706.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 26 2018
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