OFFSET
0,5
COMMENTS
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<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).
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 24 2011
STATUS
approved