A352027 a(n) = binomial(2*n-1,n) - n*(n-1) - 1.

0, 0, 3, 22, 105, 431, 1673, 6378, 24237, 92287, 352605, 1351945, 5200143, 20058117, 77558549, 300539954, 1166802837, 4537567343, 17672631557, 68923264029, 269128936799, 1052049481397, 4116715363293, 16123801840997, 63205303218275, 247959266473401, 973469712823353

Offset: 1

Author

Enrique Navarrete, Feb 28 2022

Keywords

• nonn

Comments

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 2 boxes remaining empty.

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

Examples

			a(4)=22 since 4 can be written as 3+1+0+0, 0+3+0+1, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 4+0+0+0 (4 such compositions).
		

Crossrefs

Cf. A010763, A350653.

Programs

Mathematica

Table[Binomial[2n-1,n]-n(n-1)-1,{n,40}] (* Harvey P. Dale, Dec 03 2022 *)

PARI

a(n) = binomial(2*n-1,n) - n*(n-1) - 1; \\ Michel Marcus, Apr 12 2022

Formula

G.f.: 2*x/(4*x - 1 + sqrt(1 - 4*x)) - (1 - 2*x + 3*x^2)/(1 - x)^3. - Stefano Spezia, Mar 01 2022

D-finite with recurrence n*a(n) +2*(-3*n+2)*a(n-1) +(9*n-14)*a(n-2) +2*(-2*n+5)*a(n-3) +2*(-9*n+22)=0. - R. J. Mathar, Jan 25 2023