A324804 a(n) is the number of endofunctions on a set of size n with preimage constraint {0, 1, 2, 3, 4}.

1, 1, 4, 27, 256, 3120, 46470, 817950, 16612120, 382367160, 9836517600, 279684716850, 8709747354000, 294818964039600, 10777792243818600, 423193629950091000, 17762853608696196000, 793668469023770340000, 37611450798744238416000, 1884235285123539720372000

Offset: 0

Author

Benjamin Otto, Mar 15 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 4 elements. Equivalently, it is the number of n-letter words from an n-letter alphabet such that no letter appears more than 4 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386 (first 64 terms from Benjamin Otto)
• B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.

Crossrefs

Column k=4 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(4, n))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..21);  # 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[4, n]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, May 31 2019, after Alois P. Heinz *)

Python

# print first num_entries entries in the sequence import math, sympy; x=sympy.symbols('x') k=4; 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_4(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_4 * n^(-1/2) * r_4^-n.