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

1,3

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

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

Sequence in context: A122889 A014280 A073431 * A337717 A003317 A145062

Adjacent sequences: A273870 A273871 A273872 * A273874 A273875 A273876

Keyword

nonn

Author

Gus Wiseman, Jun 01 2016

Status

approved