

A325207


a(n) is the number of labeled rooted trees on a set of size n where each node has at most 8 neighbors that are further away from the root than the node itself.


1



0, 1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 999999990, 25937423490, 743008289364, 23298080054964, 793714478374818, 29192909282466930, 1152920554828545360, 48661137306426044400, 2185908358103092063584, 104127157513055758393026, 5242868049702388548952080
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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 nonroot 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 8 entries.


LINKS

Table of n, a(n) for n=0..20.
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.3 and 7.8.


FORMULA

a(n) = (n1)! * [x^(n1)] e_8(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_8 * n^(3/2) * r_8^n.


PROG

(Python)
# print first num_entries entries in the sequence
import math, sympy; x=sympy.symbols('x')
k=8; 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_entries1):
...curr_pow=(curr_pow*eP).expand()
...r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)


CROSSREFS

Column k=8 of A325201; see that entry for sequences related to other preimage constraints constructions.
Sequence in context: A036777 A325205 A325206 * A325208 A055860 A152917
Adjacent sequences: A325204 A325205 A325206 * A325208 A325209 A325210


KEYWORD

easy,nonn


AUTHOR

Benjamin Otto, Apr 11 2019


STATUS

approved



