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

%I A196842 #18 Sep 16 2023 10:44:02
%S A196842 1,1,1,1,3,2,1,7,14,8,1,12,49,78,40,1,18,121,372,508,240,1,25,247,
%T A196842 1219,3112,3796,1680,1,33,447,3195,12864,28692,32048,13440,1,42,744,
%U A196842 7218,41619,144468,290276,301872,120960,1,52,1164,14658,113799,560658,1734956,3204632,3139680,1209600
%N A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).
%C A196842 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.
%F A196842 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.
%F A196842 a(n,k) = 0 if n<k, = |s(n+1,n+1-k)| if 0<=n<3, and= sum((-3)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=3, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).
%e A196842 n\k   0    1    2     3      4      5     6       7  ...
%e A196842 0:    1
%e A196842 1:    1    1
%e A196842 2:    1    3    2
%e A196842 3:    1    7   14     8
%e A196842 4:    1   12   49    78     40
%e A196842 5:    1   18  121   372    508    240
%e A196842 6:    1   25  247  1219   3112   3796   1680
%e A196842 7:    1   33  447  3195  12864  28692  32048  13440
%e A196842 ...
%e A196842 a(1,0) = a_0(1):= 1, a(1,1) = a_1(1)= 1.
%e A196842 a(3,2) = a_2(1,2,4) = 1*2 + 1*4 + 2*4 = 14.
%e A196842 a(3,2) = 1*|s(5,3)| - 3*|s(5,4)| + 9*|s(5,5)| = 1*35-3*10+9*1 = 14.
%p A196842 A196842 := proc(n,k)
%p A196842     if n = 1 and k =1 then
%p A196842         1 ;
%p A196842     else
%p A196842         add( abs( combinat[stirling1](n+2,n+2-k+m))*(-3)^m,m=0..k) ;
%p A196842     end if;
%p A196842 end proc: # _R. J. Mathar_, Oct 01 2016
%t A196842 a[n_, k_] := If[n == 1 && k == 1, 1, Sum[(-3)^m Abs[StirlingS1[n + 2, n + 2 - k + m]], {m, 0, k}]];
%t A196842 Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 16 2023, after _R. J. Mathar_ *)
%Y A196842 Cf. A094638, A145324,|A123319|, A196841, A055998 (column k=1), A002301 (diagonal), A277132 (subdiagonal).
%K A196842 nonn,easy,tabl
%O A196842 0,5
%A A196842 _Wolfdieter Lang_, Oct 24 2011