login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..19.

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 = [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)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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