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A024198 4th elementary symmetric function of the first n+3 odd positive integers.

%I A024198 #29 Jul 09 2025 22:19:33
%S A024198 105,1689,12139,57379,208054,626934,1646778,3889578,8439783,17085783,
%T A024198 32645613,59394517,103613692,174281212,283927812,449681892,694529781,
%U A024198 1048818981,1552033791,2254874391,3221672146,4533175570,6289743070
%N A024198 4th elementary symmetric function of the first n+3 odd positive integers.
%H A024198 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%H A024198 Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%F A024198 a(n) = n*(n+1)*(n+2)*(n+3)*(15*n^4+150*n^3+515*n^2+672*n+223)/360.
%F A024198 G.f.: -x*(x^4+112*x^3+718*x^2+744*x+105) / (x-1)^9. - _Colin Barker_, Aug 15 2014
%F A024198 a(n) = A000332(n+3) * (15*n^4+150*n^3+515*n^2+672*n+223)/15 . - _R. J. Mathar_, Oct 01 2016
%F A024198 a(n) = A(n+4, n-1), n >= 1 (fifth diagonal). See a crossref. below. - _Wolfdieter Lang_, Jul 21 2017
%F A024198 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - _Wesley Ivan Hurt_, Jul 09 2025
%t A024198 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{105,1689,12139,57379,208054,626934,1646778,3889578,8439783},30] (* _Harvey P. Dale_, May 28 2018 *)
%o A024198 (PARI) Vec(-x*(x^4+112*x^3+718*x^2+744*x+105)/(x-1)^9 + O(x^100)) \\ _Colin Barker_, Aug 15 2014
%Y A024198 From _Johannes W. Meijer_, Jun 08 2009: (Start)
%Y A024198 Equals fifth right hand column of A028338 triangle.
%Y A024198 Equals fifth left hand column of A109692 triangle.
%Y A024198 Equals fifth right hand column of A161198 triangle divided by 2^m.
%Y A024198 (End)
%K A024198 nonn,easy
%O A024198 1,1
%A A024198 _Clark Kimberling_