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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 (list; graph; refs; listen; history; text; internal format)
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 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) = (n-1)! * [x^(n-1)] 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_entries-1):

...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

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Last modified February 25 14:40 EST 2021. Contains 341609 sequences. (Running on oeis4.)