A024184 -id:A024184 - cp's OEIS frontend

A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

5, 19, 47, 95, 170, 280, 434, 642, 915, 1265, 1705, 2249, 2912, 3710, 4660, 5780, 7089, 8607, 10355, 12355, 14630, 17204, 20102, 23350, 26975, 31005, 35469, 40397, 45820, 51770, 58280, 65384, 73117, 81515, 90615, 100455, 111074, 122512, 134810, 148010

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A115127.

Partial sums of A005586.

Table[n(n+1)(n^2+13n+46)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{5,19,47,95,170},40] (* Harvey P. Dale, Apr 28 2014 *)
CoefficientList[Series[(5 - 6 x + 2 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n) = n*(n+1)*(n^2+13*n+46)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n)=A115127(n+2, 3), n>=2.
a(n) = n*(n+1)*(n^2+13n+46)/24 =a(n-1)+A005586(n). - Henry Bottomley, Oct 25 2001
G.f.: x*(5-6*x+2*x^2)/(1-x)^5.
a(n) = floor(A024184(n)/A055998(n+2)). -  R. J. Mathar, Sep 15 2009
a(1)=5, a(2)=19, a(3)=47, a(4)=95, a(5)=170, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 28 2014

A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

Original entry on oeis.org

1, 1, 3, 1, 7, 12, 1, 12, 47, 60, 1, 18, 119, 342, 360, 1, 25, 245, 1175, 2754, 2520, 1, 33, 445, 3135, 12154, 24552, 20160, 1, 42, 742, 7140, 40369, 133938, 241128, 181440, 1, 52, 1162, 14560, 111769, 537628, 1580508, 2592720, 1814400, 1, 63, 1734, 27342, 271929, 1767087, 7494416, 19978308, 30334320, 19958400

Offset: 0

Wolfdieter Lang, Oct 26 2011

For the symmetric functions a_k see a comment in A196841.
In general the triangle S_{i,j}(n,k), n>=k>=0, 1<=i=i as a_k(1,2,...,i-1,i+1,...,j-1,j+1,...,n+2).
  a_0():=1. The present triangle is S_{1,2}(n,k) (no 1 and 2 admitted).

			n\k  0   1    2     3     4       5       6       7  ...
0:   1
1:   1   3
2:   1   7   12
3:   1  12   47    60
4:   1  18  119   342   360
5:   1  25  245  1175  2754    2520
6:   1  33  445  3135 12154   24552   20160
7:   1  42  742  7140 40369  133938  241128  181440
...
a(3,2) = a_2(3,4,5) = 3*4+3*5+4*5 = 47.
a(3,2) = 1*(|s(6,4)| - (1*14 + 2*13)) + 2*(|s(6,6)| -(1*0+2*0)) = 85 - 40 + 2(1-0) = 47.
a(4,3) =  a_3(3,4,5,6) = 3*4*5+3*4*6+3*5*6+4*5*6 = 342.
a(4,3) = 1*(|s(7,4)| - (1*155 + 2*137)) + 2*(|s(7,6)| - (1*1 + 2*1)) = 735-429+2*(21-3) = 342.

Cf. A196841, A048994, A145324, A001710 (diagonal), A001711 (1st subdiagonal), A001712 (2nd subdiagonal), A055998 (k=1), A024183 (k=2), A024184 (k=3), A024185 (k=4).

a(n,k) = 0 if n=0, k=0,...,n, with the elementary symmetric function a_k (see the comment above).

a(n,k) = sum(2^k*( |s(n+3,n+3-k+2*p)| -(S_1(n+1,k-1-2*p) +2*S_2(n+1,k-1-2*p))), p=0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_1(n,k)= A145324(n+1,k+1) and S_2(n,k) = A196841(n,k).

A024193 Integer part of (3rd elementary symmetric function of S(n))/(2nd elementary symmetric of S(n)), where S(n) = {3,4, ..., n+4}.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 15, 19, 23, 27, 32, 36, 42, 47, 53, 59, 66, 73, 80, 88, 95, 104, 112, 121, 130, 140, 150, 160, 171, 182, 193, 204, 216, 228, 241, 254, 267, 281, 295, 309, 323, 338, 353

Offset: 1

Clark Kimberling

nonn

			For n=2, the 3rd elementary symmetric function of (3,4,5,6) is 3*4*5 + 3*4*6 + 3*5*6 + 4*5*6 = 342, and the 2nd elementary symmetric function of (3,4,5,6) is 3*4 + 3*5 + 3*6 + 4*5 + 4*6 + 5*6 = 119.  So 342/119 = 2.8739..., and a(2) = 2. - _Michael B. Porter_, May 05 2018

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A024184, A024183.

Table[Floor[1/2 x (7 + x) (46 + 13 x + x^2)/(144 + 41 x + 3 x^2)], {x, 43}] (* Ivan Neretin, May 02 2018 *)

a(n) = floor(A024184(n)/A024183(n+1)). - R. J. Mathar, Sep 23 2016

a(n) = floor(1/2 n (7 + n) (46 + 13 n + n^2)/(144 + 41 n + 3 n^2)). - Ivan Neretin, May 17 2018

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A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

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A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

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A024193 Integer part of (3rd elementary symmetric function of S(n))/(2nd elementary symmetric of S(n)), where S(n) = {3,4, ..., n+4}.

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