A090448 Fourth column (m=3) of triangle A090447.

9, 96, 500, 1800, 5145, 12544, 27216, 54000, 99825, 174240, 290004, 463736, 716625, 1075200, 1572160, 2247264, 3148281, 4332000, 5865300, 7826280, 10305449, 13406976, 17250000, 21970000, 27720225, 34673184, 43022196, 52983000, 64795425, 78725120, 95065344

Offset: 3

Author

Wolfdieter Lang, Dec 23 2003

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000292, A000330, A090447.

Programs

Maple

seq(mul(binomial(n,k),k=1..3),n=3..30); # Zerinvary Lajos, Dec 13 2007

Mathematica

a[n_] := Product[Binomial[n, k], {k, 0, 3}]; Array[a, 30, 3] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{9,96,500,1800,5145,12544,27216},40] (* Harvey P. Dale, Jul 18 2025 *)

Formula

a(n) = A090447(n,3).

a(n) = (n^3*(n-1)^2*(n-2)^1)/(1!*2!*3!) for n >= 3.

From Colin Barker, Jan 21 2013: (Start)

a(n) = (n^6-4*n^5+5*n^4-2*n^3)/12.

G.f.: -x^3*(x^3+17*x^2+33*x+9)/(x-1)^7. (End)

a(n) = A000330(n-1)^2 - A000292(n-1)^2. - J. M. Bergot, May 02 2014

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, May 04 2021

From Amiram Eldar, Sep 08 2022: (Start)

Sum_{n>=3} 1/a(n) = 207/4 - 9*Pi^2/2 - 6*zeta(3).

Sum_{n>=3} (-1)^(n+1)/a(n) = 165/4 - Pi^2/4 - 48*log(2) - 9*zeta(3)/2. (End)