%I #57 Jul 08 2019 16:56:39

%S 3,23,86,230,505,973,1708,2796,4335,6435,9218,12818,17381,23065,30040,

%T 38488,48603,60591,74670,91070,110033,131813,156676,184900,216775,

%U 252603,292698,337386,387005,441905,502448,569008,641971,721735,808710,903318

%N a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers.

%H Bruno Berselli, <a href="/A024196/b024196.txt">Table of n, a(n) for n = 1..1000</a>

%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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n*(n+1)*(3*n^2+5*n+1)/6.

%F From _Bruno Berselli_, Mar 13 2012: (Start)

%F G.f.: x*(3 + 8*x + x^2)/(1 - x)^5.

%F a(n) = Sum_{i=1..n} (n+1-i)*((n+1)^2-i).

%F a(n) = n*A016061(n) - Sum_{i=0..n-1} A016061(i). (End)

%F a(n) - a(n-1) = A099721(n). Partial sums of A099721.- _Philippe Deléham_, May 07 2012

%F a(n) = Sum_{i=1..n} ((2*i-1)*Sum_{j=i..n} (2*j+1)) = 1*(3+5+...2*n+1) + 3*(5+7+...+2*n+1) + ... + (2*n-1)*(2*n+1). - _J. M. Bergot_, Apr 21 2017

%F a(n) = A028338(n+1, n-1), n >= 1, (third diagonal). See the crossref. below. _Wolfdieter Lang_, Jul 21 2017

%F a(n) = (A000583(n+1) - A000447(n+1))/2. - _J. M. Bergot_, Feb 13 2018

%e a(8) = 8*80+7*79+6*78+5*77+4*76+3*75+2*74+1*73 = 2796. - _Bruno Berselli_, Mar 13 2012

%p seq(n*(n+1)*(3*n^2+5*n+1)/6,n=1..25); # _Muniru A Asiru_, Feb 13 2018

%t f[k_] := 2 k - 1; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[2, t[n]]

%t Table[a[n], {n, 2, 50}] (* A024196 *)

%t (* _Clark Kimberling_, Dec 31 2011 *)

%t Table[(n(n+1)(3n^2+5n+1))/6,{n,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{3,23,86,230,505},50] (* _Harvey P. Dale_, Jul 08 2019 *)

%o (GAP) List([1..36], n -> n*(n+1)*(3*n^2+5*n+1)/6); # _Muniru A Asiru_, Feb 13 2018

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

%Y Equals third right hand column of A028338 triangle.

%Y Equals third left hand column of A109692 triangle.

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

%Y (End)

%Y Cf. A016061.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_