A325207 - OEIS

%I #8 Nov 16 2024 17:36:59

%S 0,1,2,9,64,625,7776,117649,2097152,43046721,999999990,25937423490,

%T 743008289364,23298080054964,793714478374818,29192909282466930,

%U 1152920554828545360,48661137306426044400,2185908358103092063584,104127157513055758393026,5242868049702388548952080

%N 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.

%C 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.

%H 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 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.

%o (Python)

%o # print first num_entries entries in the sequence

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

%o k=8; num_entries = 64

%o P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [0,1]; curr_pow = eP

%o for term in range(1,num_entries-1):

%o     curr_pow=(curr_pow*eP).expand()

%o     r.append(curr_pow.coeff(x**term)*math.factorial(term))

%o print(r)

%Y Column k=8 of A325201; see that entry for sequences related to other preimage constraints constructions.

%K easy,nonn

%O 0,3

%A _Benjamin Otto_, Apr 11 2019