OFFSET
1,9
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,k) is the number of endofunctions on a set of size n with exactly one cyclic point and such that each preimage has at most k entries.
LINKS
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.3 and 7.8.
FORMULA
A(n,k) = (n-1)! * [x^(n-1)] e_k(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. When k>1, the link above yields explicit constants c_k, r_k so that the columns are asymptotically c_k * n^(-3/2) * r_k^-n. Stirling's approximation gives column k=1, and column k=0 is 0.
EXAMPLE
Array begins:
0 0 0 0 0 ...
0 1 1 1 1 ...
0 2 2 2 2 ...
0 6 9 9 9 ...
0 24 60 64 64 ...
0 120 540 620 625 ...
0 720 6120 7620 7770 ...
0 5040 83790 113610 117390 ...
0 40320 1345680 1992480 2088520 ...
0 362880 24811920 40194000 42771960 ...
0 3628800 516650400 916927200 991090800 ...
0 39916800 11992503600 23341071600 25635767850 ...
...
MATHEMATICA
e[k_][x_] := Sum[x^j/j!, {j, 0, k}];
A[0, _] = A[_, 0] = 0; A[n_, k_] := (n-1)! Coefficient[e[k][x]^n, x, n-1];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] (* Jean-François Alcover, Jul 06 2019 *)
PROG
(Python)
# print first num_entries entries in column k
import math, sympy; x=sympy.symbols('x')
k=5; 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
KEYWORD
AUTHOR
Benjamin Otto, Apr 08 2019
STATUS
approved