A365330 - OEIS

%I #12 Sep 30 2023 21:41:30

%S 0,0,0,6,48,600,8640,131040,2257920,42819840,885427200,19918483200,

%T 483791616000,12622171161600,352200296448000,10466625641472000,

%U 330077933273088000,11010660024139776000,387369218691366912000,14335266857678807040000,556691771706962411520000

%N Expansion of e.g.f. x^3/(1-x-x^2-x^3)^2.

%C 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 3 elements from a block.

%F a(n) = A000142(n)*A073778(n+1).

%e a(6)=8640 since the ways to partition [6] into blocks of size at most 3, order the blocks, order the elements within each block, and select 3 elements from a block are the following:

%e   (i) 123,4,5,6: 2880 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 2880 ways;

%e   (ii) 123,45,6: 4320 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 4320 ways;

%e   (iii) 123,456: 720 such orderings, 2 ways to choose three elements (from one of the two blocks with 3 elements), hence 1440 ways.

%t With[{m = 20}, Range[0, m]! * CoefficientList[Series[x^3/(1 - x - x^2 - x^3)^2, {x, 0, m}], x]] (* _Amiram Eldar_, Sep 02 2023 *)

%Y Cf. A000073, A000142, A189886, A364324, A364422.

%K nonn

%O 0,4

%A _Enrique Navarrete_, Sep 01 2023