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A324809
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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, 9}.
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1
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1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 9999999990, 285311669390, 8916100350828, 302875100019492, 11112006413890382, 437893865348970030, 18446742559675475760, 827240169494482480880, 39346402337538654701772, 1978419291074273862219834
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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, 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.
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LINKS
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FORMULA
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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.
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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(9, 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[9, 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=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)
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CROSSREFS
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Column k=9 of A306800; see that entry for sequences related to other preimage constraints constructions.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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