A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).

1, 1, 1, 1, 3, 2, 1, 7, 14, 8, 1, 12, 49, 78, 40, 1, 18, 121, 372, 508, 240, 1, 25, 247, 1219, 3112, 3796, 1680, 1, 33, 447, 3195, 12864, 28692, 32048, 13440, 1, 42, 744, 7218, 41619, 144468, 290276, 301872, 120960, 1, 52, 1164, 14658, 113799, 560658, 1734956, 3204632, 3139680, 1209600

Offset: 0

Author

Wolfdieter Lang, Oct 24 2011

Keywords

• nonn
• easy
• tabl

Comments

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[1]=1, x[2]=2, and x[j]=j+1 for j=3,...,n. This is the triangle S_3(n,k), n>=0, k=0..n. The first three rows coincide with those of triangle A094638.

Examples

			n\k   0    1    2     3      4      5     6       7  ...
0:    1
1:    1    1
2:    1    3    2
3:    1    7   14     8
4:    1   12   49    78     40
5:    1   18  121   372    508    240
6:    1   25  247  1219   3112   3796   1680
7:    1   33  447  3195  12864  28692  32048  13440
...
a(1,0) = a_0(1):= 1, a(1,1) = a_1(1)= 1.
a(3,2) = a_2(1,2,4) = 1*2 + 1*4 + 2*4 = 14.
a(3,2) = 1*|s(5,3)| - 3*|s(5,4)| + 9*|s(5,5)| = 1*35-3*10+9*1 = 14.
		

Crossrefs

Cf. A094638, A145324,|A123319|, A196841, A055998 (column k=1), A002301 (diagonal), A277132 (subdiagonal).

Programs

Maple

A196842 := proc(n,k)
    if n = 1 and k =1 then
        1 ;
    else
        add( abs( combinat[stirling1](n+2,n+2-k+m))*(-3)^m,m=0..k) ;
    end if;
end proc: # R. J. Mathar, Oct 01 2016

Mathematica

a[n_, k_] := If[n == 1 && k == 1, 1, Sum[(-3)^m Abs[StirlingS1[n + 2, n + 2 - k + m]], {m, 0, k}]];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2023, after R. J. Mathar *)

Formula

a(n,k) = a_k(1,2,..,n) if 0<=n<3, and a_k(1,2,4,5,...,n+1) if n>=3, with the elementary symmetric functions a_k defined in a comment to A196841.

a(n,k) = 0 if n=3, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).