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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}.

%I A324809 #15 Nov 21 2024 20:12:51
%S A324809 1,1,4,27,256,3125,46656,823543,16777216,387420489,9999999990,
%T A324809 285311669390,8916100350828,302875100019492,11112006413890382,
%U A324809 437893865348970030,18446742559675475760,827240169494482480880,39346402337538654701772,1978419291074273862219834
%N 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}.
%C A324809 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.
%C A324809 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.
%H A324809 B. Otto, <a href="https://arxiv.org/abs/1903.00542">Coalescence under Preimage Constraints</a>, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.
%F A324809 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.
%p A324809 b:= proc(n, i) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
%p A324809       add(b(n-j, i-1)*binomial(n, j), j=0..min(9, n))))
%p A324809     end:
%p A324809 a:= n-> b(n$2):
%p A324809 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 01 2019
%t A324809  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]}]]];
%t A324809 a[n_] := b[n, n];
%t A324809 a /@ Range[0, 20] (* _Jean-François Alcover_, Mar 01 2020, after _Alois P. Heinz_ *)
%o A324809 (Python)
%o A324809 # print first num_entries entries in the sequence
%o A324809 import math, sympy; x=sympy.symbols('x')
%o A324809 k=9; num_entries = 64
%o A324809 P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [1]; curr_pow = 1
%o A324809 for term in range(1, num_entries):
%o A324809    curr_pow=(curr_pow*eP).expand()
%o A324809    r.append(curr_pow.coeff(x**term)*math.factorial(term))
%o A324809 print(r)
%Y A324809 Column k=9 of A306800; see that entry for sequences related to other preimage constraints constructions.
%K A324809 easy,nonn
%O A324809 0,3
%A A324809 _Benjamin Otto_, Mar 25 2019