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A325205 a(n) is the number of labeled rooted trees on a set of size n where each node has at most 6 neighbors that are further away from the root than the node itself. 1
0, 1, 2, 9, 64, 625, 7776, 117649, 2097144, 43046136, 999970020, 25936053990, 742947675624, 23295384644532, 793591829158128, 29187143427692250, 1152639088016576160, 48646833059722978080, 2185150741063924810176, 104085328898784937079376, 5240461483486301616704160 (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 6 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_6(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_6 * n^(-3/2) * r_6^-n.

PROG

(Python)

# print first num_entries entries in the sequence

import math, sympy; x=sympy.symbols('x')

k=6; 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=6 of A325201; see that entry for sequences related to other preimage constraints constructions.

Sequence in context: A274395 A036776 A036777 * A325206 A325207 A325208

Adjacent sequences:  A325202 A325203 A325204 * A325206 A325207 A325208

KEYWORD

easy,nonn

AUTHOR

Benjamin Otto, Apr 11 2019

STATUS

approved

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Last modified March 8 11:48 EST 2021. Contains 341948 sequences. (Running on oeis4.)