A358341 Expansion of e.g.f. (exp(x)-1)*(exp(x)-x)*(exp(x)-x^2/2).

0, 1, 3, 7, 31, 96, 314, 1072, 3693, 12556, 41800, 136236, 435923, 1374088, 4280358, 13211704, 40492633, 123440724, 374774660, 1134346228, 3425446335, 10326139696, 31088511778, 93507747360, 281053811141, 844319049436, 2535473717184, 7611873731452, 22847398782763, 68567563479576

Offset: 0

Author

Enrique Navarrete, Feb 22 2023

Keywords

• nonn
• easy

Comments

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first set is not empty, the second set cannot have a single element, and the third set cannot have two elements.

Examples

			The 31 set partitions for n=4 are the following:
{1,2,3,4}, { }, { } (1 of these);
{1,2,3}, { }, {4};  (4 of these);
{1,2}, {3,4}, { };  (6 of these);
{1}, {2,3,4}, { };  (4 of these);
{1}, {2,3}, {4};   (12 of these);
{1}, { }, {2,3,4};  (4 of these).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (13,-72,222,-417,489,-350,140,-24).

Crossrefs

Cf. A360586.

Formula

a(n) = 3^n - 2^n - n*(2^(n-1)-1) - binomial(n,2)*(2^(n-2)-1) + 3*binomial(n,3) except at n=3.