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A300443
Number of binary enriched p-trees of weight n.
11
1, 1, 2, 3, 8, 15, 41, 96, 288, 724, 2142, 5838, 17720, 49871, 151846, 440915, 1363821, 4019460, 12460721, 37374098, 116809752, 353904962, 1109745666, 3396806188, 10712261952, 33006706419, 104357272687, 323794643722, 1027723460639, 3204413808420, 10193485256501
OFFSET
0,3
COMMENTS
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.
LINKS
FORMULA
a(n) = 1 + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).
EXAMPLE
The a(4) = 8 binary enriched p-trees: 4, (31), (22), ((21)1), ((11)2), (2(11)), (((11)1)1), ((11)(11)).
MAPLE
a:= proc(n) option remember;
1+add(a(j)*a(n-j), j=1..n/2)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Mar 06 2018
MATHEMATICA
j[n_]:=j[n]=1+Sum[Times@@j/@y, {y, Select[IntegerPartitions[n], Length[#]===2&]}];
Array[j, 40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n-j], {j, 1, n/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 05 2018
STATUS
approved