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A325206 a(n) is the number of labeled rooted trees on a set of size n where each node has at most 7 neighbors that are further away from the root than the node itself.

%I A325206 #8 Dec 21 2024 17:51:31
%S A325206 0,1,2,9,64,625,7776,117649,2097152,43046712,999999180,25937373990,
%T A325206 743005653984,23297946618804,793707788417544,29192570114517810,
%U A325206 1152902963147295360,48660197610533102880,2185856466420637543104,104124189019562479248624,5242691958381764070687360
%N A325206 a(n) is the number of labeled rooted trees on a set of size n where each node has at most 7 neighbors that are further away from the root than the node itself.
%C A325206 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 7 entries.
%H A325206 B. Otto, <a href="https://arxiv.org/abs/1903.00542">Coalescence under Preimage Constraints</a>, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.3 and 7.8.
%F A325206 a(n) = (n-1)! * [x^(n-1)] e_7(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_7 * n^(-3/2) * r_7^-n.
%o A325206 (Python)
%o A325206 # print first num_entries entries in the sequence
%o A325206 import math, sympy; x=sympy.symbols('x')
%o A325206 k=7; num_entries = 64
%o A325206 P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [0,1]; curr_pow = eP
%o A325206 for term in range(1,num_entries-1):
%o A325206     curr_pow=(curr_pow*eP).expand()
%o A325206     r.append(curr_pow.coeff(x**term)*math.factorial(term))
%o A325206 print(r)
%Y A325206 Column k=7 of A325201; see that entry for sequences related to other preimage constraints constructions.
%K A325206 easy,nonn
%O A325206 0,3
%A A325206 _Benjamin Otto_, Apr 11 2019