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

0,4

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).

Links

Table of n, a(n) for n=0..39.

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

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

Example

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).

Mathematica

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

Prog

(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

Crossrefs

Cf. A006000, A361099.

Sequence in context: A011855 A173031 A350413 * A004404 A319552 A201189

Adjacent sequences: A361471 A361472 A361473 * A361475 A361476 A361477

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 13 2023

Status

approved