A361099 a(n) = n + 2*binomial(n,2) + 3*binomial(n,3) + 4*binomial(n,4).

0, 1, 4, 12, 32, 75, 156, 294, 512, 837, 1300, 1936, 2784, 3887, 5292, 7050, 9216, 11849, 15012, 18772, 23200, 28371, 34364, 41262, 49152, 58125, 68276, 79704, 92512, 106807, 122700, 140306, 159744, 181137, 204612, 230300, 258336, 288859, 322012, 357942, 396800, 438741

Offset

0,3

Comments

a(n) is the number of ordered set partitions of an n-set into 2 sets such that the first set has either 3, 2, 1 or no elements, the second set has no restrictions, and an element is selected from the second set.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

E.g.f.: (1 + x + x^2/2 + x^3/6)*x*exp(x).

From Stefano Spezia, Mar 04 2023: (Start)

O.g.f.: x*(1 - x + 2*x^2 + 2*x^3)/(1 - x)^5.

a(n) = A000290(n) + A004320(n-2). (End)

Example

The 294 set partitions for n=7 are the following (where the element selected from the second set is in parentheses):
{ }, {(1),2,3,4,5,6,7}  (7 of these);
{1}, {(2),3,4,5,6,7}   (42 of these);
{1,2}, {(3),4,5,6,7}   (105 of these);
{1,2,3}, {(4),5,6,7}   (140 of these).

Mathematica

Table[n^2*(n*(n - 3) + 8)/6, {n, 0, 50}] (* Paolo Xausa, Jun 10 2024 *)

Prog

(Python)
def A361099(n): return n**2*(n*(n - 3) + 8)//6 # Chai Wah Wu, Mar 24 2023

Crossrefs

Cf. A000290, A004320.

Sequence in context: A133212 A233447 A127811 * A138517 A001934 A004403

Adjacent sequences: A361096 A361097 A361098 * A361100 A361101 A361102

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 01 2023

Status

approved