OFFSET
0,3
COMMENTS
A preimage constraint on a function is a set of nonnegative integers such that the size of the inverse image of any element is one of the values in that set. View a labeled rooted tree as an endofunction on the set {1,2,...,n} by sending every non-root node to its neighbor that is closer to the root and sending the root to itself. Thus, a(n) is the number of endofunctions on a set of size n with exactly one cyclic point and such that each preimage has at most 9 entries.
LINKS
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.3 and 7.8.
FORMULA
a(n) = (n-1)! * [x^(n-1)] e_9(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. The link above yields explicit constants c_k, r_k so that the columns are asymptotically c_9 * n^(-3/2) * r_9^-n.
MATHEMATICA
e[k_][x_] := Sum[x^j/j!, {j, 0, k}];
a[0] = 0; a[n_] := (n - 1)! Coefficient[e[9][x]^n, x, n - 1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 06 2019 *)
PROG
(Python) # print first num_entries entries in the sequence
import math, sympy; x=sympy.symbols('x')
k=9; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [0, 1]; curr_pow = eP
for term in range(1, num_entries-1):
...curr_pow=(curr_pow*eP).expand()
...r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benjamin Otto, Jul 05 2019
STATUS
approved