A301753 - OEIS

%I #10 Aug 29 2018 02:52:34

%S 1,1,2,3,6,9,16,25,43,66,108,166,269,408,643,975,1517,2277,3497,5223,

%T 7936,11803,17736,26219,39174,57594,85299,124957,183987,268158,392685,

%U 569987,830282,1200843,1740422,2507823,3620550,5197885,7472229,10694865,15319700

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

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

%H Andrew Howroyd, <a href="/A301753/b301753.txt">Table of n, a(n) for n = 1..500</a>

%F O.g.f.: x * Product_{n > 0} 1/(1 - A000009(n-1) x^n).

%e The a(7) = 16 rooted twice-partitions:

%e (5), (32), (41),

%e (2)(2), (3)(1), (4)(), (21)(1), (31)(),

%e (1)(1)(1), (2)(1)(), (3)()(), (21)()(),

%e (1)(1)()(), (2)()()(),

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

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

%t nn=50;

%t ser=x*Product[1/(1-PartitionsQ[n-1]x^n),{n,nn}];

%t Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

%o (PARI) seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(1/prod(k=1, n-1, 1 - u[k]*x^k + O(x^n)))} \\ _Andrew Howroyd_, Aug 29 2018

%Y Cf. A002865, A032305, A063834, A093637, A270995, A281113, A296119, A301422, A301462, A301467, A301480, A301706, A301751.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 26 2018