A366942 -id:A366942 - cp's OEIS frontend

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

Enrique Navarrete, Oct 29 2023

nonn

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.

			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.

Cf. A000262, A189886, A364324, A366942.

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

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

A366957 Expansion of e.g.f. 1/(1-x^2-3*x^3).

Original entry on oeis.org

1, 0, 2, 18, 24, 720, 7200, 45360, 1128960, 14152320, 199584000, 4909766400, 82388275200, 1793381990400, 47163455539200, 1051370191872000, 29396519792640000, 863253387988992000, 24437860434763776000, 807966756915462144000, 27000346486744350720000

Offset: 0

Enrique Navarrete, Oct 30 2023

nonn

For n>0, 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 2 elements from each block.
E.g.: a(6)=7200 since we have the following cases:
 12,34,56: 720 such orderings, 1 way to choose two elements from each block;
 123,456: 720 such orderings, 3*3 ways to choose two elements from each block;
so 720*1 + 720*9 = 7200 ways.

Cf. A000073, A189886, A364324, A366942, A366950.

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

cp's OEIS Frontend

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

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