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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}. 1
 1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 9999999990, 285311669390, 8916100350828, 302875100019492, 11112006413890382, 437893865348970030, 18446742559675475760, 827240169494482480880, 39346402337538654701772, 1978419291074273862219834 (list; graph; refs; listen; history; text; internal format)
 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 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 = ; 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 Column k=9 of A306800; see that entry for sequences related to other preimage constraints constructions. Sequence in context: A117280 A067040 A070271 * A245414 A000312 A177885 Adjacent sequences:  A324806 A324807 A324808 * A324810 A324811 A324812 KEYWORD easy,nonn AUTHOR Benjamin Otto, Mar 25 2019 STATUS approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)