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A324808
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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}.
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2
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1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420480, 9999999090, 285311608890, 8916096836988, 302874907278372, 11111996007290178, 437893300069054830, 18446711285575475760, 827238394513348062960, 39346298554172667026112, 1978413024254468818876002
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OFFSET
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0,3
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COMMENTS
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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, a(n) is the number of endofunctions on a set of size n such that each preimage has at most 8 elements. Equivalently, it is the number of n-letter words from an n-letter alphabet such that no letter appears more than 8 times.
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LINKS
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FORMULA
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a(n) = n! * [x^n] e_8(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_8 * n^(-1/2) * r_8^-n.
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MAPLE
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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(8, n))))
end:
a:= n-> b(n$2):
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MATHEMATICA
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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[8, n]}]]];
a[n_] := b[n, n];
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PROG
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(Python)
# print first num_entries entries in the sequence
import math, sympy; x=sympy.symbols('x')
k=8; 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)
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CROSSREFS
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Column k=8 of A306800; see that entry for sequences related to other preimage constraints constructions.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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