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%I #35 Sep 08 2022 08:44:48
%S 60,342,1175,3135,7140,14560,27342,48150,80520,129030,199485,299117,
%T 436800,623280,871420,1196460,1616292,2151750,2826915,3669435,4710860,
%U 5986992,7538250,9410050,11653200,14324310,17486217,21208425,25567560
%N Third elementary symmetric function of 3,4,...,n+4.
%H Ivan Neretin, <a href="/A024184/b024184.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48.
%F 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
%F 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
%F G.f.: x*(8*x^3-41*x^2+78*x-60) / (x-1)^7. - _Colin Barker_, Aug 15 2014
%F a(n) = A000292(n) * (n+7)*(n^2+13*n+46)/8. - _R. J. Mathar_, Oct 01 2016
%p seq(n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48,n=1..40); # _Muniru A Asiru_, May 19 2018
%t 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 *)
%o (PARI) Vec(x*(8*x^3-41*x^2+78*x-60)/(x-1)^7 + O(x^100)) \\ _Colin Barker_, Aug 15 2014
%o (Magma) [n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48: n in [1..40]]; // _Vincenzo Librandi_, May 03 2018
%o (GAP) List([1..30],n->n*(n+1)*(n+2)*(n+7)*(n^2+13*n+46)/48); # _Muniru A Asiru_, May 19 2018
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
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