OFFSET
0,3
COMMENTS
An enriched p-tree of weight n is either (case 1) the number n itself, or (case 2) a sequence of two or more enriched p-trees, having a weakly decreasing sequence of weights summing to n.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1588
FORMULA
O.g.f.: (1/(1-x) + Product_{i>0} 1/(1-a(i)*x^i))/2.
EXAMPLE
The a(4) = 12 enriched p-trees are:
4,
(31), ((21)1), (((11)1)1), ((111)1),
(22), (2(11)), ((11)2), ((11)(11)),
(211), ((11)11),
(1111).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 07 2017
MATHEMATICA
a[n_]:=a[n]=1+Sum[Times@@a/@y, {y, Rest[IntegerPartitions[n]]}];
Array[a, 20]
(* Alternative: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
If[i<1, 0, b[n, i-1] + a[i] b[n-i, Min[n-i, i]]]];
a[n_] := If[n == 0, 1, 1 + b[n, n-1]];
a /@ Range[0, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)
PROG
(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 07 2017
STATUS
approved
