OFFSET
1,3
COMMENTS
A strict tree t is either (case 1) a positive integer t = n, or (case 2) a set t = {t1, t2, ..., tk} of two or more strict trees (i.e. branches) with distinct weights, where the weight of a strict tree in the second case is the sum of the weights of its branches; hence the multiset of weights is a strict integer partition of n. For example, {{{{{2,1},1},2},3},{4,{2,1}},{2,1},1} is a strict tree of weight 20.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2000
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47(1995), 1219-1239
Gus Wiseman, Comcategories and Multiorders, (pdf version)
Gus Wiseman, All strict trees n=1..6.
FORMULA
Sum_{g(t)=y} (-1)^{d(t)} = mu(|y|<={y_1,...,y_k}), where mu is the Mobius function of the multiorder of integer partitions, g(t) is the multiset of leaves of a strict tree t, and d(t) is the number of branchings.
Strict trees are closely related to the coefficients appearing in a(i) = Sum_y c(y_1)*...*c(y_k) where Sum_i c(i)*x^i = Prod_i (1 + a(i)*x^i). The latter identity is the formal power product expansion (PPE) of an (ordinary) generating function.
EXAMPLE
a(6) = 12: {6, (51), (42), ((41)1), (321), ((31)2), ((32)1), (((31)1)1), ((21)21), (((21)1)2), (((21)2)1), ((((21)1)1)1)}.
MAPLE
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1)+b(n-i, min(n-i, i-1))*a(i)))
end:
a:= n-> 1+b(n, n-1):
seq(a(n), n=1..35); # Alois P. Heinz, Jun 02 2016
MATHEMATICA
STE[n_Integer?Positive]:=STE[n]=1+Plus@@Map[Function[ptn, Times@@STE/@ptn], Select[IntegerPartitions[n], And[Length[#]>1, UnsameQ@@#]&]];
Array[STE, 30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[i(i + 1)/2 < n, 0,
If[n == 0, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]] a[i]]];
a[n_] := If[n == 0, 1, 1 + b[n, n - 1]];
a /@ Range[35] (* 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(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); v} \\ Andrew Howroyd, Aug 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 01 2016
STATUS
approved