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Number of ways to choose a rooted partition of each part in a constant rooted partition of n.
1

%I #9 Aug 27 2018 01:52:21

%S 1,1,2,3,5,6,13,12,26,31,57,43,150,78,224,293,484,232,1190,386,2260,

%T 2087,2558,1003,11154,4701,7889,13597,30041,3719,83248,5605,95006,

%U 84486,63506,251487,654394,17978,169864,490741,2290336,37339,4079503,53175,3979370

%N Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

%C A rooted partition of n is an integer partition of n - 1.

%H Andrew Howroyd, <a href="/A301761/b301761.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{d | n-1} A000041((n-1)/d-1)^d for n > 1. - _Andrew Howroyd_, Aug 26 2018

%e The a(7) = 13 rooted twice-partitions:

%e (5), (41), (32), (311), (221), (2111), (11111),

%e (2)(2), (2)(11), (11)(2), (11)(11),

%e (1)(1)(1),

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

%t Table[Sum[PartitionsP[n/d-1]^d,{d,Divisors[n]}],{n,50}]

%o (PARI) a(n)=if(n==1, 1, sumdiv(n-1, d, numbpart((n-1)/d-1)^d)) \\ _Andrew Howroyd_, Aug 26 2018

%Y Cf. A000005, A000041, A002865, A018818, A063834, A093637, A127524, A260685, A271619, A279787, A301422, A301462, A301467, A301480, A301706.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 26 2018