A361474 a(n) = 1*binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5).

0, 0, 1, 6, 24, 80, 225, 546, 1176, 2304, 4185, 7150, 11616, 18096, 27209, 39690, 56400, 78336, 106641, 142614, 187720, 243600, 312081, 395186, 495144, 614400, 755625, 921726, 1115856, 1341424, 1602105, 1901850, 2244896, 2635776, 3079329, 3580710, 4145400, 4779216, 5488321, 6279234

Offset: 0

Author

Enrique Navarrete, Mar 13 2023

Keywords

• nonn
• easy

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 two elements are selected from the second set.

Note the coefficients 1,3,6,10 in a(n) are triangular numbers (in accordance with the selection of two elements from the second set).

Examples

			The 546 set partitions for n=7 are the following (where the 2 elements selected from the second set are in parentheses):
   { }, {(1),(2),3,4,5,6,7}  (21 of these);
   {1}, {(2),(3),4,5,6,7}    (105 of these);
   {1,2}, {(3),(4),5,6,7}    (210 of these);
   {1,2,3}, {(4),(5),6,7}    (210 of these).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006000, A361099.

Programs

Mathematica

a[n_] := Total[Binomial[n, Range[2, 5]]*{1, 3, 6, 10}]; Array[a, 40, 0] (* Amiram Eldar, Mar 28 2023 *)

PARI

a(n) = binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5); \\ Michel Marcus, Mar 16 2023

Python

def A361474(n): return n*(n*(n*(n*(n - 7) + 23) - 29) + 12)//12 # Chai Wah Wu, Apr 16 2023

Formula

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

O.g.f.: x^2*(1 + 3*x^2 + 6*x^3)/(1 - x)^6. - Stefano Spezia, Mar 16 2023