A352405 a(n) = binomial(n,2)*(binomial(n-1,2) + 2).

0, 2, 9, 30, 80, 180, 357, 644, 1080, 1710, 2585, 3762, 5304, 7280, 9765, 12840, 16592, 21114, 26505, 32870, 40320, 48972, 58949, 70380, 83400, 98150, 114777, 133434, 154280, 177480, 203205, 231632, 262944, 297330, 334985, 376110, 420912, 469604, 522405, 579540, 641240, 707742, 779289, 856130

Offset: 1

Author

Enrique Navarrete, Mar 14 2022

Keywords

• nonn
• easy

Comments

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with either 1 or 2 boxes remaining empty.

a(n) is also the number of weak compositions of n into n parts that contain either one or two 0's.

a(n)+1 is the number of ways to place n indistinguishable balls into n distinguishable boxes with at most 2 boxes remaining empty (just add the case of no empty boxes in which we place exactly one ball in one box).

Examples

			a(4)=30 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); 2+1+1+0 (12 such compositions).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A010763, A350653.

Programs

Mathematica

a[n_] := Binomial[n, 2] * (Binomial[n - 1, 2] + 2); Array[a, 50] (* Amiram Eldar, Mar 15 2022 *)

Formula

G.f.: x^2*(2 - x + 5*x^2)/(1 - x)^5. - Stefano Spezia, Mar 15 2022