A324809 a(n) is the number of endofunctions on a set of size n with preimage constraint {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 9999999990, 285311669390, 8916100350828, 302875100019492, 11112006413890382, 437893865348970030, 18446742559675475760, 827240169494482480880, 39346402337538654701772, 1978419291074273862219834

Offset: 0

Author

Benjamin Otto, Mar 25 2019

Keywords

• easy
• nonn

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.

Crossrefs

Column k=9 of A306800; see that entry for sequences related to other preimage constraints constructions.

Programs

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 *)

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)

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.