A301480 - OEIS

%I #11 Aug 29 2018 02:52:26

%S 1,1,2,4,8,15,30,54,103,186,345,606,1115,1936,3466,6046,10630,18257,

%T 31927,54393,93894,159631,272155,458891,779375,1305801,2196009,

%U 3667242,6130066,10170745,16923127,27942148,46211977,76039205,125094369,204952168,335924597

%N Number of rooted twice-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.

%H Andrew Howroyd, <a href="/A301480/b301480.txt">Table of n, a(n) for n = 1..500</a>

%F O.g.f.: x * Product_{n > 0} 1/(1 - P(n-1) x^n) where P = A000041.

%e The a(5) = 8 rooted twice-partitions: ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()).

%e The a(6) = 15 rooted twice-partitions:

%e (4), (31), (22), (211), (1111),

%e (3)(), (21)(), (111)(), (2)(1), (11)(1),

%e (2)()(), (11)()(), (1)(1)(),

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

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

%t nn=30;

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

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

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

%Y Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 22 2018