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 A024196 a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers. 8
 3, 23, 86, 230, 505, 973, 1708, 2796, 4335, 6435, 9218, 12818, 17381, 23065, 30040, 38488, 48603, 60591, 74670, 91070, 110033, 131813, 156676, 184900, 216775, 252603, 292698, 337386, 387005, 441905, 502448, 569008, 641971, 721735, 808710, 903318 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Bruno Berselli, Table of n, a(n) for n = 1..1000 Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017. Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = n*(n+1)*(3*n^2+5*n+1)/6. From Bruno Berselli, Mar 13 2012: (Start) G.f.: x*(3 + 8*x + x^2)/(1 - x)^5. a(n) = Sum_{i=1..n} (n+1-i)*((n+1)^2-i). a(n) = n*A016061(n) - Sum_{i=0..n-1} A016061(i). (End) a(n) - a(n-1) = A099721(n). Partial sums of A099721.- Philippe Deléham, May 07 2012 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 a(n) = A028338(n+1, n-1), n >= 1, (third diagonal). See the crossref. below. Wolfdieter Lang, Jul 21 2017 a(n) = (A000583(n+1) - A000447(n+1))/2. - J. M. Bergot, Feb 13 2018 EXAMPLE a(8) = 8*80+7*79+6*78+5*77+4*76+3*75+2*74+1*73 = 2796. - Bruno Berselli, Mar 13 2012 MAPLE seq(n*(n+1)*(3*n^2+5*n+1)/6, n=1..25); # Muniru A Asiru, Feb 13 2018 MATHEMATICA f[k_] := 2 k - 1; t[n_] := Table[f[k], {k, 1, n}] a[n_] := SymmetricPolynomial[2, t[n]] Table[a[n], {n, 2, 50}]  (* A024196 *) (* Clark Kimberling, Dec 31 2011 *) 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 *) PROG (GAP) List([1..36], n -> n*(n+1)*(3*n^2+5*n+1)/6); # Muniru A Asiru, Feb 13 2018 CROSSREFS From Johannes W. Meijer, Jun 08 2009: (Start) Equals third right hand column of A028338 triangle. Equals third left hand column of A109692 triangle. Equals third right hand column of A161198 triangle divided by 2^m. (End) Cf. A016061. Sequence in context: A201482 A032017 A197453 * A196339 A196318 A213846 Adjacent sequences:  A024193 A024194 A024195 * A024197 A024198 A024199 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 8 21:32 EDT 2021. Contains 343668 sequences. (Running on oeis4.)