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A289501 Number of enriched p-trees of weight n. 83
1, 1, 2, 4, 12, 32, 112, 352, 1296, 4448, 16640, 59968, 231168, 856960, 3334400, 12679424, 49991424, 192890880, 767229952, 2998427648, 12015527936, 47438950400, 191117033472, 760625733632, 3082675150848, 12346305839104, 50223511928832, 202359539335168 (list; graph; refs; listen; history; text; internal format)
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
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]
(* Second program: *)
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
Sequence in context: A216819 A216820 A148194 * A274961 A027695 A148195
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 07 2017
STATUS
approved

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Last modified March 19 02:49 EDT 2024. Contains 370952 sequences. (Running on oeis4.)