A350653 a(n) is the number of weak compositions of n into n-1 parts in which at least one part is zero.

0, 2, 12, 52, 205, 786, 2996, 11432, 43749, 167950, 646635, 2496132, 9657687, 37442146, 145422660, 565722704, 2203961413, 8597496582, 33578000591, 131282408380, 513791607399, 2012616400058, 7890371113927, 30957699535752, 121548660036275

Offset: 2

Author

Enrique Navarrete, Jan 09 2022

Keywords

• nonn
• easy

Examples

			a(5)=52 since 5 can be written as 5+0+0+0 (4 such compositions); 4+1+0+0 (12 such compositions); 3+2+0+0 (12 such compositions); 3+1+1+0 (12 such compositions); 2+2+1+0 (12 such compositions). All these weak compositions contain at least one zero.
		

Crossrefs

Cf. A001791, A010763.

Programs

Mathematica

a[n_] := Binomial[2*n - 2, n] - n + 1; Array[a, 25, 2] (* Amiram Eldar, Jan 10 2022 *)

Formula

a(n) = binomial(2*n-2,n) - (n-1) = A001791(n-1) -n+1.

G.f.: 4*x^2/((1 - sqrt(1 - 4*x))^2*sqrt(1 - 4*x)) - (1 - 2*x + 2*x^2)/(1 - x)^2. - Stefano Spezia, Jan 10 2022

D-finite with recurrence +n*(11*n-38)*a(n) -(n-1)*(73*n-244)*a(n-1) +2*(67*n^2-364*n+492)*a(n-2) -4*(9*n-22)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Mar 06 2022