cp's OEIS Frontend

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

%I A366950 #25 Dec 14 2023 09:33:33
%S A366950 1,0,2,18,12,360,3360,7560,183120,1814400,8195040,184615200,
%T A366950 1976546880,14166472320,310589959680,3634245014400,36092331475200,
%U A366950 787170153369600,10123771065408000,127736406404006400,2807613032557132800,39732753299855616000
%N A366950 Expansion of e.g.f. exp(x^2+3*x^3).
%C A366950 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.
%C A366950 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.
%F A366950 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
%e A366950 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:
%e A366950 1) 12,34,56: 120 ways to order elements in unordered blocks, 1 way to choose 2 elements from each block, hence 120 ways;
%e A366950 2) 123,456: 360 ways to order elements in unordered blocks, 3*3 ways to choose 2 elements from each block, hence 3240 ways.
%t A366950 With[{m = 21}, Range[0, m]! * CoefficientList[Series[Exp[x^2 + 3*x^3], {x, 0, m}], x]] (* _Amiram Eldar_, Oct 30 2023 *)
%Y A366950 Cf. A000262, A189886, A364324, A366942.
%K A366950 nonn
%O A366950 0,3
%A A366950 _Enrique Navarrete_, Oct 29 2023