cp's OEIS Frontend

A024183 Second elementary symmetric function of 3,4,...,n+3.

%I A024183 #38 Aug 15 2025 21:08:30
%S A024183 12,47,119,245,445,742,1162,1734,2490,3465,4697,6227,8099,10360,13060,
%T A024183 16252,19992,24339,29355,35105,41657,49082,57454,66850,77350,89037,
%U A024183 101997,116319,132095,149420,168392,189112,211684,236215,262815,291597,322677,356174
%N A024183 Second elementary symmetric function of 3,4,...,n+3.
%H A024183 Ivan Neretin, <a href="/A024183/b024183.txt">Table of n, a(n) for n = 1..10000</a>
%H A024183 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A024183 a(n) = n*(n+1)*(3*n^2 + 35*n + 106)/24.
%F A024183 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-2,3), for n >= 3. - _Milan Janjic_, Dec 20 2008
%F A024183 From _Colin Barker_, Aug 15 2014: (Start)
%F A024183 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A024183 G.f.: -x*(4*x^2-13*x+12)/(x-1)^5. (End)
%F A024183 E.g.f.: exp(x)*x*(6 + x)*(48 + 38*x + 3*x^2)/24. - _Elmo R. Oliveira_, Aug 15 2025
%p A024183 seq(n*(n+1)*(3*n^2+35*n+106)/24,n=1..40); # _Muniru A Asiru_, May 19 2018
%t A024183 f[k_] := k + 2; t[n_] := Table[f[k], {k, 1, n}]
%t A024183 a[n_] := SymmetricPolynomial[2, t[n]]
%t A024183 Table[a[n], {n, 2, 30}]  (* A024183 *)
%t A024183 (* _Clark Kimberling_, Dec 31 2011 *)
%t A024183 LinearRecurrence[{5, -10, 10, -5, 1}, {12, 47, 119, 245, 445}, 40] (* _Vincenzo Librandi_, May 03 2018 *)
%o A024183 (PARI) Vec(-x*(4*x^2-13*x+12)/(x-1)^5 + O(x^100)) \\ _Colin Barker_, Aug 15 2014
%o A024183 (Magma) [n*(n+1)*(3*n^2+35*n+106)/24: n in [1..40]]; // _Vincenzo Librandi_, May 03 2018
%o A024183 (GAP) List([1..40],n->n*(n+1)*(3*n^2+35*n+106)/24); # _Muniru A Asiru_, May 19 2018
%K A024183 nonn,easy
%O A024183 1,1
%A A024183 _Clark Kimberling_