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A289501
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Number of enriched p-trees of weight n.
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83
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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O.g.f.: (1/(1-x) + Product_{i>0} 1/(1-a(i)*x^i))/2.
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EXAMPLE
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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).
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MAPLE
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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)):
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MATHEMATICA
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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]];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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