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Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.
2

%I #8 Aug 27 2018 01:52:37

%S 1,1,2,3,4,4,8,5,8,13,14,5,32,7,20,64,26,6,92,7,126,199,22,5,352,252,

%T 41,581,394,7,1832,9,292,2119,31,3216,4946,10,40,8413,7708,9,20656,9,

%U 2324,53546,24,5,70040,16395,59361,131204,9503,7,266780,178180,82086

%N Number of ways to choose a constant 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="/A301763/b301763.txt">Table of n, a(n) for n = 1..1000</a>

%e The a(7) = 8 rooted twice-partitions: (5), (11111), (2)(2), (2)(11), (11)(2), (11)(11), (1)(1)(1), ()()()()()().

%e The a(15) = 20 rooted twice-partitions:

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

%e (1)(1)(1)(1)(1)(1)(1), (111111)(111111), (1111111111111),

%e (111111)(222), (222)(111111), (222)(222),

%e (111111)(33), (222)(33), (33)(111111), (33)(222), (33)(33),

%e (111111)(6), (222)(6), (33)(6), (6)(111111), (6)(222), (6)(33), (6)(6),

%e (13).

%t Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]

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

%Y Cf. A000005, A002865, A047968, A063834, A093637, A127524, A295924, A300383, A301422, A301462, A301467, A301480, A301706.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 26 2018