A357340 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268438(n - k, j).

1, -1, 1, 2, -2, 1, 0, 12, -3, 1, -56, -120, 28, -4, 1, 0, 1680, -450, 50, -5, 1, 15840, -30240, 10416, -1080, 78, -6, 1, 0, 665280, -317520, 33712, -2100, 112, -7, 1, -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1

Offset: 0

Author

Peter Luschny, Sep 25 2022

Keywords

• sign
• tabl

Examples

			Triangle T(n, k) starts:
[0]         1;
[1]        -1,         1;
[2]         2,        -2,        1;
[3]         0,        12,       -3,        1;
[4]       -56,      -120,       28,       -4,     1;
[5]         0,      1680,     -450,       50,    -5,     1;
[6]     15840,    -30240,    10416,    -1080,    78,    -6,   1;
[7]         0,    665280,  -317520,    33712, -2100,   112,  -7,  1;
[8] -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1;
		

Crossrefs

Cf. A357341 (alternating row sums), A264437, A268438, A357339.

Programs

Maple

A357340 := proc(n, k) local u; u := n - k; (2*u)!*add(binomial(-n, j) * j! *
  add((-1)^(j+m)*binomial(u+j, u+m)*abs(Stirling1(u+m, m)), m=0..j)/(u +j)!, j=0..u) end: seq(print(seq(A357340(n, k), k=0..n)), n=0..8);

SageMath

# using function A268438
def A357340(n, k):
    return sum(binomial(-n, i) * A268438(n - k, i) for i in range(n - k + 1))
for n in range(10): print([A357340(n, k) for k in range(n + 1)])