A196845 - OEIS

%I #12 Oct 01 2016 10:34:41

%S 1,1,3,1,7,12,1,12,47,60,1,18,119,342,360,1,25,245,1175,2754,2520,1,

%T 33,445,3135,12154,24552,20160,1,42,742,7140,40369,133938,241128,

%U 181440,1,52,1162,14560,111769,537628,1580508,2592720,1814400,1,63,1734,27342,271929,1767087,7494416,19978308,30334320,19958400

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

%C For the symmetric functions a_k see a comment in A196841.

%C In general the triangle S_{i,j}(n,k), n>=k>=0, 1<=i<j<=n+2 is defined for n<i as a_k(1,2,...,n), and for n>=i as a_k(1,2,...,i-1,i+1,...,j-1,j+1,...,n+2).

%C   a_0():=1. The present triangle is S_{1,2}(n,k) (no 1 and 2 admitted).

%F a(n,k) = 0 if n<k, a(n,k) = a_k(3,4,...,n+2), n>=0, k=0,...,n, with the elementary symmetric function a_k (see the comment above).

%F 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).

%e n\k  0   1    2     3     4       5       6       7  ...

%e 0:   1

%e 1:   1   3

%e 2:   1   7   12

%e 3:   1  12   47    60

%e 4:   1  18  119   342   360

%e 5:   1  25  245  1175  2754    2520

%e 6:   1  33  445  3135 12154   24552   20160

%e 7:   1  42  742  7140 40369  133938  241128  181440

%e ...

%e a(3,2) = a_2(3,4,5) = 3*4+3*5+4*5 = 47.

%e 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.

%e a(4,3) =  a_3(3,4,5,6) = 3*4*5+3*4*6+3*5*6+4*5*6 = 342.

%e 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.

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

%K nonn,easy,tabl

%O 0,3

%A _Wolfdieter Lang_, Oct 26 2011