cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Wolfdieter Lang, Oct 26 2011

Keywords

Comments

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).

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

Clark Kimberling

Keywords

Examples

			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

Links

Crossrefs

Cf. A024184, A024183.

Programs

  • Mathematica
    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 *)

Formula

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

A024194 [ (4th elementary symmetric function of S(n))/(2nd elementary symmetric of S(n)) ], where S(n) = {3,4, ..., n+5}.

Original entry on oeis.org

3, 11, 27, 54, 96, 156, 240, 353, 501, 688, 923, 1212, 1564, 1985, 2485, 3074, 3760, 4554, 5466, 6508, 7692, 9028, 10532, 12214, 14090, 16172, 18477, 21018, 23813, 26876, 30224, 33876, 37847, 42157, 46825, 51868, 57308, 63164, 69457, 76208, 83438, 91170
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A024185, A024183.

Programs

  • Mathematica
    Table[Floor[1/240 x (1 + x) (53848 + 26922 x + 5165 x^2 + 450 x^3 + 15 x^4)/(188 + 47 x + 3 x^2)], {x, 42}] (* Ivan Neretin, May 02 2018 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.