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A372245 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)).

%I A372245 #19 May 19 2024 12:01:49
%S A372245 1,-1,1,3,-1,2,-1,7,0,6,27,11,26,12,24,119,151,120,150,120,120,1203,
%T A372245 1139,1202,1140,1200,1080,720,11759,11887,11760,11886,11760,11760,
%U A372245 10080,5040,136587,136331,136586,136332,136584,136080,131040,100800,40320,1771559,1772071,1771560,1772070
%N A372245 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 A372245 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 A372245 T(n, 0) = A344037(n).
%F A372245 T(n, 1) = A052841(n) - A344037(n).
%F A372245 T(n, 2) = A344037(n) - 2*A052841(n) + A000670(n).
%e A372245 Triangle T(n, k) starts:
%e A372245 [0]  1;
%e A372245 [1] -1,      1;
%e A372245 [2]  3,     -1,      2;
%e A372245 [3] -1,      7,      0,      6;
%e A372245 [4]  27,     11,     26,     12,     24;
%e A372245 [5]  119,    151,    120,    150,    120,    120;
%e A372245 [6]  1203,   1139,   1202,   1140,   1200,   1080,   720;
%e A372245 [7]  11759,  11887,  11760,  11886,  11760,  11760,  10080,  5040;
%e A372245 [8]  136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320;
%o A372245 (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 A372245 Cf. A000670, A052841, A084416, A341091, A344037.
%K A372245 sign,easy,tabl
%O A372245 0,4
%A A372245 _Thomas Scheuerle_, Apr 26 2024