A024196 a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers.

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

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

Clark Kimberling

Status

approved