A024184 Third elementary symmetric function of 3,4,...,n+4.

60, 342, 1175, 3135, 7140, 14560, 27342, 48150, 80520, 129030, 199485, 299117, 436800, 623280, 871420, 1196460, 1616292, 2151750, 2826915, 3669435, 4710860, 5986992, 7538250, 9410050, 11653200, 14324310, 17486217, 21208425, 25567560

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Programs

GAP

List([1..30],n->n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48); # Muniru A Asiru, May 19 2018

Magma

[n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48: n in [1..40]]; // Vincenzo Librandi, May 03 2018

Maple

seq(n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48,n=1..40); # Muniru A Asiru, May 19 2018

Mathematica

Table[n(n+1)(n+2)(n+7)(n^2+13n+46)/48,{n,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{60,342,1175,3135,7140,14560,27342},30] (* Harvey P. Dale, Oct 31 2011 *)

PARI

Vec(x*(8*x^3-41*x^2+78*x-60)/(x-1)^7 + O(x^100)) \\ Colin Barker, Aug 15 2014

Formula

a(n) = n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48.

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n-2) = -f(n,n-3,3), for n >= 3. - Milan Janjic, Dec 20 2008

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), with a(1)=60, a(2)=342, a(3)=1175, a(4)=3135, a(5)=7140, a(6)=14560, a(7)=27342. - Harvey P. Dale, Oct 31 2011

G.f.: x*(8*x^3-41*x^2+78*x-60) / (x-1)^7. - Colin Barker, Aug 15 2014

a(n) = A000292(n) * (n+7)*(n^2+13*n+46)/8. - R. J. Mathar, Oct 01 2016