A325208 - OEIS

%I #14 Dec 22 2024 09:08:14

%S 0,1,2,9,64,625,7776,117649,2097152,43046721,1000000000,25937424590,

%T 743008369224,23298084997044,793714764270428,29192925433321650,

%U 1152921466989795360,48661189511753527280,2185911410555033096364,104127340753401006230046,5242879377215160617336400

%N a(n) is the number of labeled rooted trees on a set of size n where each node has at most 9 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 9 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_9(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_9 * n^(-3/2) * r_9^-n.

%t e[k_][x_] := Sum[x^j/j!, {j, 0, k}];

%t a[0] = 0; a[n_] := (n - 1)! Coefficient[e[9][x]^n, x, n - 1];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 06 2019 *)

%o (Python) # print first num_entries entries in the sequence

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

%o k=9; 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=9 of A325201; see that entry for sequences related to other preimage constraints constructions.

%K easy,nonn

%O 0,3

%A _Benjamin Otto_, Jul 05 2019