A273873 Number of strict trees of weight n.

1, 1, 2, 3, 6, 12, 28, 65, 166, 412, 1076, 2806, 7524, 20020, 54744, 148417, 410078, 1126732, 3144500, 8728570, 24555900, 68713420, 194469616, 548088278, 1559301428, 4418131108, 12628267512, 35957541462, 103150588492, 294924202032, 848878072440, 2435729999665

Offset: 1

Author

Gus Wiseman, Jun 01 2016

Keywords

• nonn

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.

Examples

			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)}.
		

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.

Crossrefs

Cf. A196545 (weakly ordered plane trees); A220418, A220420 (power product expansions); A271619, A063834 (twice partitioned numbers), A289501.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2 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 *)

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

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.