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%I #20 May 12 2021 12:07:28
%S 1,1,2,3,8,15,41,96,288,724,2142,5838,17720,49871,151846,440915,
%T 1363821,4019460,12460721,37374098,116809752,353904962,1109745666,
%U 3396806188,10712261952,33006706419,104357272687,323794643722,1027723460639,3204413808420,10193485256501
%N Number of binary enriched p-trees of weight n.
%C A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.
%H Alois P. Heinz, <a href="/A300443/b300443.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 1 + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).
%e The a(4) = 8 binary enriched p-trees: 4, (31), (22), ((21)1), ((11)2), (2(11)), (((11)1)1), ((11)(11)).
%p a:= proc(n) option remember;
%p 1+add(a(j)*a(n-j), j=1..n/2)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Mar 06 2018
%t j[n_]:=j[n]=1+Sum[Times@@j/@y,{y,Select[IntegerPartitions[n],Length[#]===2&]}];
%t Array[j,40]
%t (* Second program: *)
%t a[n_] := a[n] = 1 + Sum[a[j]*a[n-j], {j, 1, n/2}];
%t a /@ Range[0, 40] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)
%o (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, n\2, v[k]*v[n-k])); concat([1], v)} \\ _Andrew Howroyd_, Aug 26 2018
%Y Cf. A000992, A001190, A063834, A196545, A273873, A289501, A300354, A300439, A300442.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 05 2018
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