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4th elementary symmetric function of the first n+3 odd positive integers.
5

%I #27 May 28 2018 11:36:41

%S 105,1689,12139,57379,208054,626934,1646778,3889578,8439783,17085783,

%T 32645613,59394517,103613692,174281212,283927812,449681892,694529781,

%U 1048818981,1552033791,2254874391,3221672146,4533175570,6289743070

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

%H <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 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 a(n) = n*(n+1)*(n+2)*(n+3)*(15*n^4+150*n^3+515*n^2+672*n+223)/360.

%F G.f.: -x*(x^4+112*x^3+718*x^2+744*x+105) / (x-1)^9. - _Colin Barker_, Aug 15 2014

%F 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 a(n) = A(n+4, n-1), n >= 1 (fifth diagonal). See a crossref. below. - _Wolfdieter Lang_, Jul 21 2017

%t 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 (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 From _Johannes W. Meijer_, Jun 08 2009: (Start)

%Y Equals fifth right hand column of A028338 triangle.

%Y Equals fifth left hand column of A109692 triangle.

%Y Equals fifth right hand column of A161198 triangle divided by 2^m.

%Y (End)

%K nonn,easy

%O 1,1

%A _Clark Kimberling_