A371003 a(n) = binomial(2*n-1,n) - binomial(n,2)*(binomial(n-1,2) + 2) - 1.

0, 0, 0, 4, 45, 281, 1358, 5790, 23229, 90667, 350130, 1348315, 5194995, 20051019, 77548994, 300527354, 1166786517, 4537546535, 17672605394, 68923231539, 269128896899, 1052049432887, 4116715304850, 16123801771169, 63205303135475, 247959266375901, 973469712709278

Offset: 1

Author

Enrique Navarrete, Mar 07 2024

Keywords

• nonn

Comments

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 3 boxes remaining empty.

a(n) is also the number of weak compositions of n into n parts in which at least three parts are zero.

Examples

			a(5)=45 since 5 can be written as 5+0+0+0+0, 0+5+0+0+0, etc. (5 such compositions); 4+1+0+0+0 (20 such compositions); 3+2+0+0+0 (20 such compositions).
		

Crossrefs

Cf. A010763, A352027, A359435.

Programs

Mathematica

Table[Binomial[2n-1,n]-Binomial[n,2]*(Binomial[n-1,2]+2)-1,{n,27}] (* James C. McMahon, Mar 08 2024 *)

Python

from math import comb
def A371003(n): return comb((n<<1)-1,n)-n-((m:=(n-1)**2)*(m+3)>>2) # Chai Wah Wu, Mar 29 2024