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
Alois P. Heinz, Table of n, a(n) for n = 0..1000
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 05 2018
STATUS
approved