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A273873 Number of strict trees of weight n. 67
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 (list; graph; refs; listen; history; text; internal format)
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
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47(1995), 1219-1239
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 01 2016
STATUS
approved

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Last modified April 17 04:44 EDT 2024. Contains 371756 sequences. (Running on oeis4.)