A366950 Expansion of e.g.f. exp(x^2+3*x^3).

1, 0, 2, 18, 12, 360, 3360, 7560, 183120, 1814400, 8195040, 184615200, 1976546880, 14166472320, 310589959680, 3634245014400, 36092331475200, 787170153369600, 10123771065408000, 127736406404006400, 2807613032557132800, 39732753299855616000

Offset

0,3

Comments

For n>0, a(n) is the number of ways to partition [n] into unordered blocks of size at most 3, order the elements within each block, and choose 2 elements from each block.

For example, a(2)=2 since the blocks with ordered elements are 12 and 21 and there is only one way to choose 2 elements from each block.

Links

Table of n, a(n) for n=0..21.

Formula

a(n) ~ 3^(2*n/3 - 1/2) * n^(2*n/3) * exp(4/729 - 2*3^(-11/3)*n^(1/3) + 3^(-4/3)*n^(2/3) - 2*n/3). - Vaclav Kotesovec, Nov 02 2023

Example

a(6)=3360 since the number of ways to partition [6] into unordered blocks of size at most 3, order the elements within each block, and select 2 elements from each block are the following:
1) 12,34,56: 120 ways to order elements in unordered blocks, 1 way to choose 2 elements from each block, hence 120 ways;
2) 123,456: 360 ways to order elements in unordered blocks, 3*3 ways to choose 2 elements from each block, hence 3240 ways.

Mathematica

With[{m = 21}, Range[0, m]! * CoefficientList[Series[Exp[x^2 + 3*x^3], {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)

Crossrefs

Cf. A000262, A189886, A364324, A366942.

Sequence in context: A174708 A328648 A359197 * A144158 A083502 A076378

Adjacent sequences: A366947 A366948 A366949 * A366951 A366952 A366953

Keyword

nonn

Author

Enrique Navarrete, Oct 29 2023

Status

approved