A264429 Triangle read by rows, inverse Bell transform of Bell numbers.

1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, 0, 7, -6, 1, 0, -5, -10, 25, -10, 1, 0, 18, -20, -75, 65, -15, 1, 0, -7, 231, 70, -315, 140, -21, 1, 0, -338, -840, 1064, 945, -980, 266, -28, 1, 0, 2215, -1278, -8918, 1512, 4935, -2520, 462, -36, 1

Offset: 0

Author

Peter Luschny, Nov 13 2015

Keywords

• sign
• tabl

Examples

			[ 1 ]
[ 0,     1 ]
[ 0,    -1,     1 ]
[ 0,     1,    -3,     1 ]
[ 0,     0,     7,    -6,     1 ]
[ 0,    -5,   -10,    25,   -10,     1 ]
[ 0,    18,   -20,   -75,    65,   -15,     1 ]
[ 0,    -7,   231,    70,  -315,   140,   -21,     1 ]
[ 0,  -338,  -840,  1064,   945,  -980,   266,   -28,     1 ]
[ 0,  2215, -1278, -8918,  1512,  4935, -2520,   462,   -36,   1 ]
		

Links

• Peter Luschny, The Bell transform

Crossrefs

Cf. A000110, A264428.

Programs

Mathematica

rows = 10;
M = Table[BellY[n, k, BellB[Range[0, rows-1]]],{n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
A264429 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Sage

# uses[bell_transform from A264428]
def inverse_bell_transform(dim, L):
    M = matrix(ZZ, dim)
    for n in range(dim):
        row = bell_transform(n, L)
        for k in (0..n): M[n,k] = row[k]
    return M.inverse()
def A264429_matrix(dim):
    uno = [1]*dim
    bell_numbers = [sum(bell_transform(n, uno)) for n in range(dim)]
    return inverse_bell_transform(dim, bell_numbers)
A264429_matrix(10)