A366942 Expansion of e.g.f. 1/(1-x-2*x^2-3*x^3).

1, 1, 6, 48, 408, 5040, 72000, 1184400, 22619520, 482993280, 11459750400, 299495750400, 8531976499200, 263353163673600, 8754879893760000, 311808414677760000, 11845876873678848000, 478163414336864256000, 20436460099541950464000, 921972301728418676736000

Offset: 0

Author

Enrique Navarrete, Oct 29 2023

Keywords

• nonn

Comments

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, order the elements within each block, and choose 1 element from each block.

E.g.: a(4) = 408 since we have the following cases:

1,2,3,4: 24 such orderings, 1 way to choose one element from each block;

12,34: 24 such orderings, 2*2 ways to choose one element from each block;

12,3,4: 72 such orderings, 2*1*1 ways to choose one element from each block;

123,4: 48 such orderings, 3*1 ways to choose one element from each block;

so 24*1 + 24*4 + 72*2 + 48*3 = 408 ways.

Crossrefs

Cf. A000073, A000142, A052567, A052585, A101822, A189886, A364324.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*j!*j, j=1..min(3, n)))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Dec 14 2023

Mathematica

With[{m = 20}, Range[0, m]! * CoefficientList[Series[1/(1 - x - 2*x^2 - 3*x^3), {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x-2*x^2-3*x^3))) \\ Michel Marcus, Oct 30 2023

Formula

a(n) = A000142(n)*A101822(n).

a(n) = n*(a(n-1)+(n-1)*(2*a(n-2)+(n-2)*3*a(n-3))) for n>=3. - Alois P. Heinz, Dec 14 2023