A364324 - OEIS

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A364324

a(n) = n!*tribonacci(n+2).

1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120, 994291200, 20118067200, 444034483200, 10617070464000, 273391121203200, 7542665754624000, 221969877921792000, 6940528784437248000, 229781192298577920000, 8030036368187817984000, 295390797322766745600000

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OFFSET

0,3

COMMENTS

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, and order the elements within each block.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

E.g.f.: 1/(1-x-x^2-x^3).

a(n) = A000142(n) * A000073(n+2).

EXAMPLE

a(5) = 1560 since the number of ways to partition [5] into blocks of size at most 3, order the blocks, and order the elements within each block are the following:

1) 1,2,3,4,5: 120 ordered blocks; 120 ways;

2) 12,3,4,5: 240 ordered blocks; 480 ways;

3) 12,34,5: 90 ordered blocks; 360 ways;

4) 123,45: 20 ordered blocks; 240 ways;

5) 123,4,5: 60 ordered blocks; 360 ways.

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, add(

a(n-i)*binomial(n, i)*i!, i=1..min(n, 3)))

end:

seq(a(n), n=0..20); # Alois P. Heinz, Jul 18 2023

MATHEMATICA

With[{m = 21}, Range[0, m - 1]! * LinearRecurrence[{1, 1, 1}, {1, 1, 2}, m]] (* Amiram Eldar, Jul 28 2023 *)