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

2,2

Links

Table of n, a(n) for n=2..26.

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

Example

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.

Mathematica

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

Crossrefs

Cf. A001791, A010763.

Sequence in context: A176580 A179259 A261474 * A080675 A218782 A007225

Adjacent sequences: A350650 A350651 A350652 * A350654 A350655 A350656

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 09 2022

Status

approved