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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)).
1
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
0,4
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).
EXAMPLE
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;
PROG
(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))))
CROSSREFS
KEYWORD
sign,easy,tabl
AUTHOR
Thomas Scheuerle, Apr 26 2024
STATUS
approved