|
%I #9 Aug 26 2018 16:32:49
%S 1,1,1,3,6,16,47,132,410,1254,4052,12818,42783,139082,469924,1563606,
%T 5353966,18065348,62491018,213391790,743836996,2565135934,8994087070,
%U 31251762932,110245063771,385443583008,1365151504722,4800376128986,17070221456536,60289267885410
%N Number of enriched p-trees of weight n with odd leaves.
%C An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.
%H Andrew Howroyd, <a href="/A300355/b300355.txt">Table of n, a(n) for n = 0..500</a>
%F O.g.f: (1 + x/(1-x^2) + Prod_{i>0} 1/(1 - a(i)x^i))/2.
%F a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).
%e The a(5) = 16 enriched p-trees of weight with odd leaves:
%e 5,
%e ((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),
%e (3(11)), (((11)1)(11)), ((111)(11)),
%e (311), (((11)1)11), ((111)11),
%e ((11)(11)1),
%e ((11)111),
%e (11111).
%t c[n_]:=c[n]=If[EvenQ[n],0,1]+Sum[Times@@c/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
%t Table[c[n],{n,30}]
%o (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = n%2 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ _Andrew Howroyd_, Aug 26 2018
%Y Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 03 2018
|
