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 A237585 Number of structures of size n in class A = o x (o + MSET(A)) where o is a neutral structure of size 1. 1
 0, 1, 2, 3, 6, 15, 36, 94, 245, 663, 1815, 5062, 14269, 40706, 117103, 339673, 991834, 2913869, 8605576, 25536300, 76096896, 227634717, 683296679, 2057540487, 6213495745, 18813535942, 57103173296, 173710272584, 529534793886, 1617347972250, 4948744120771 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS MSET(A) is the multi-choose function: pick any number of unlabeled structures in A with repetition allowed. Interpreting the neutral structure of size 1 as a single pointer dereference, A is the class of A-pointers either to null pointers or to a multiset of unlabeled A-pointers, where the size of a pointer is the number of dereferences required to resolve the entire structure, so a null pointer has size 1 and an A-pointer to a null pointer has size 2 and an A-pointer to {A-pointer(null), A-pointer(null), A-pointer({A-pointer(null)})} has size 1+((1+1)+(1+1)+(1+(1+1)))=8. a(n) is the number of rooted trees of weight n where leaves can have either weight 1 or 2 and non-leaves have weight 1. - Andrew Howroyd, Mar 02 2020 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..200 Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009 Guy P. Srinivasan, C# program to generate sequence FORMULA G.f. A(x) satisfies: A(x) = x * (x + exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...)). - Ilya Gutkovskiy, Jun 11 2021 EXAMPLE For n = 3 the a(3)=3 pointers are the pointer to the multiset of exactly the pointer to the null pointer, the pointer to the multiset of twice the pointer to the empty multiset, and the pointer to the multiset of exactly the pointer to the multiset of the pointer to the empty multiset. From Andrew Howroyd, Mar 02 2020: (Start) The a(2) = 2 trees are: 2, (1). The a(3) = 3 trees are: (2), (11), ((1)). The a(4) = 6 trees are: ((2)), (12), (111), ((11)), (1(1)), (((1))). (End) PROG (C#) (see linked code for GetPartitions, Choose, and invoking this) private static Func A237585() {     Func A = null;     Func B = null;     Func C = null;     A = (n) => n == 0 ? 0 : B(n-1);     B = (n) => C(n) + (n == 1 ? 1 : 0);     C = (n) =>         {             if (n == 0) return 1;             long sum = 0;             foreach (var partition in GetPartitions(n))             {                 long product = 1;                 for (int k = 1; k < partition.Count; k++)                 {                     var N = A(k);                     var K = partition[k];                     product *= Choose(N + K - 1, K);                 }                 sum += product;             }             return sum;         };     return A; } (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} seq(n)={my(v=); for(n=2, n, v=concat(, EulerT(v)); v++); concat(, v)} \\ Andrew Howroyd, Mar 02 2020 CROSSREFS Sequence in context: A182240 A052102 A053561 * A147773 A006403 A129960 Adjacent sequences:  A237582 A237583 A237584 * A237586 A237587 A237588 KEYWORD nonn AUTHOR Guy P. Srinivasan, Feb 09 2014 EXTENSIONS Terms a(21) and beyond from Andrew Howroyd, Mar 02 2020 STATUS approved

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Last modified October 25 04:28 EDT 2021. Contains 348237 sequences. (Running on oeis4.)