%I #19 May 19 2024 12:01:49

%S 1,-1,1,3,-1,2,-1,7,0,6,27,11,26,12,24,119,151,120,150,120,120,1203,

%T 1139,1202,1140,1200,1080,720,11759,11887,11760,11886,11760,11760,

%U 10080,5040,136587,136331,136586,136332,136584,136080,131040,100800,40320,1771559,1772071,1771560,1772070

%N Triangular array T(n,k) read by rows: column k is the expansion of e.g.f: exp(-2*x)*(exp(x)-1)^k/(2-exp(x)).

%F T(n, k) = Sum_{m=0..n} ((-1)^(1+m+n)*binomial(k, n)*(2^(k - n) - 1)*A084416(m, k - 1)), for k > 0.

%F T(n, 0) = A344037(n).

%F T(n, 1) = A052841(n) - A344037(n).

%F T(n, 2) = A344037(n) - 2*A052841(n) + A000670(n).

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] -1, 1;

%e [2] 3, -1, 2;

%e [3] -1, 7, 0, 6;

%e [4] 27, 11, 26, 12, 24;

%e [5] 119, 151, 120, 150, 120, 120;

%e [6] 1203, 1139, 1202, 1140, 1200, 1080, 720;

%e [7] 11759, 11887, 11760, 11886, 11760, 11760, 10080, 5040;

%e [8] 136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320;

%o (PARI) T(n, k) = sum(m=0, n, ((-1)^((k > 0)+m+n)*binomial(n, m)*(2^(n-m)-(k > 0))*sum(h=max(k-1,0), m, h!*stirling(m, h, 2))))

%Y Cf. A000670, A052841, A084416, A341091, A344037.

%K sign,easy,tabl

%O 0,4

%A _Thomas Scheuerle_, Apr 26 2024