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 A289501 Number of enriched p-trees of weight n. 82
 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 Alois P. Heinz, Table of n, a(n) for n = 0..1588 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 Cf. A052337, A063834, A093637, A196545, A273873, A281145, A300660. Sequence in context: A216819 A216820 A148194 * A274961 A027695 A148195 Adjacent sequences:  A289498 A289499 A289500 * A289502 A289503 A289504 KEYWORD nonn AUTHOR Gus Wiseman, Jul 07 2017 STATUS approved

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Last modified May 22 17:42 EDT 2022. Contains 353957 sequences. (Running on oeis4.)