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)).

1, -1, 1, 3, -1, 2, -1, 7, 0, 6, 27, 11, 26, 12, 24, 119, 151, 120, 150, 120, 120, 1203, 1139, 1202, 1140, 1200, 1080, 720, 11759, 11887, 11760, 11886, 11760, 11760, 10080, 5040, 136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320, 1771559, 1772071, 1771560, 1772070

Offset: 0

Author

Thomas Scheuerle, Apr 26 2024

Keywords

• sign
• easy
• tabl

Examples

			Triangle T(n, k) starts:
[0]  1;
[1] -1,      1;
[2]  3,     -1,      2;
[3] -1,      7,      0,      6;
[4]  27,     11,     26,     12,     24;
[5]  119,    151,    120,    150,    120,    120;
[6]  1203,   1139,   1202,   1140,   1200,   1080,   720;
[7]  11759,  11887,  11760,  11886,  11760,  11760,  10080,  5040;
[8]  136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320;
		

Crossrefs

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

Programs

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))))

Formula

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.

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

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

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