A129936 a(n) = (n-2)*(n+3)*(n+2)/6.

-2, -2, 0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

Author

Roger L. Bagula, Jun 09 2007

Keywords

• easy
• sign

Comments

Essentially the same as A005586.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000292, A002378, A005586, A023444, A034856, A214292.

Programs

Maple

seq(sum(i*(k-i+1), i=1..k+2), k=0..99); # Wesley Ivan Hurt, Sep 21 2013

Mathematica

f[n_] = Binomial[n + 3, 3] - (n + 3)*(n + 2)/2; Table[f[n], {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{-2,-2,0,5},50] (* Harvey P. Dale, Jul 03 2020 *)

PARI

a(n)=(n-2)*(n+3)*(n+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = binomial(n+3,3) - (n + 3)*(n + 2)/2.

a(n) = A214292(n+2,2). - Reinhard Zumkeller, Jul 12 2012

G.f.: (x^3-4*x^2+6*x-2)/(x-1)^4. - Colin Barker, Sep 05 2012

From Wesley Ivan Hurt, Sep 21 2013: (Start)

a(n) = Sum_{i=1..n+2} i*(n-i+1).

a(n+2) = A000292(n+1) + A034856(n), n>0. (End)

From Amiram Eldar, Sep 26 2022: (Start)

Sum_{n>=3} 1/a(n) = 77/200.

Sum_{n>=3} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)

From Elmo R. Oliveira, Aug 04 2025: (Start)

E.g.f.: exp(x)*(x^3 + 6*x^2 - 12)/6.

a(n) = A023444(n)*A002378(n+2)/6.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Extensions

More terms from Wesley Ivan Hurt, Sep 21 2013