A237585 Number of structures of size n in class A = o x (o + MSET(A)) where o is a neutral structure of size 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

Offset: 0

Author

Sarah Nibs, Feb 09 2014

Keywords

• nonn

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

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009
• Sarah Nibs, C# program to generate sequence

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], EulerT(v)); v[2]++); concat([0],v)} \\ Andrew Howroyd, Mar 02 2020

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

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 02 2020