OFFSET
0,3
COMMENTS
A preimage constraint 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.
Thus, the n-th term of the sequence is the number of endofunctions on a set of size n such that each preimage has at most 9 elements. Equivalently, it is the number of n-letter words from an n-letter alphabet such that no letter appears more than 9 times.
LINKS
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.
FORMULA
a(n) = n! * [x^n] 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^(-1/2) * r_9^-n.
MAPLE
b:= proc(n, i) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
add(b(n-j, i-1)*binomial(n, j), j=0..min(9, n))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Apr 01 2019
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0 && i == 0, 1, If[i < 1, 0, Sum[b[n - j, i - 1]*Binomial[n, j], {j, 0, Min[9, n]}]]];
a[n_] := b[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)
PROG
(Python)
# print first num_entries entries in the sequence
import math, sympy; x=sympy.symbols('x')
k=9; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [1]; curr_pow = 1
for term in range(1, num_entries):
curr_pow=(curr_pow*eP).expand()
r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Benjamin Otto, Mar 25 2019
STATUS
approved