A326327 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^(-n), for m = 2, n >= 0, k >= 0; square array read by descending antidiagonals.

1, 0, 1, 0, -1, 1, 0, 5, -2, 1, 0, -61, 16, -3, 1, 0, 1385, -272, 33, -4, 1, 0, -50521, 7936, -723, 56, -5, 1, 0, 2702765, -353792, 25953, -1504, 85, -6, 1, 0, -199360981, 22368256, -1376643, 64256, -2705, 120, -7, 1, 0, 19391512145, -1903757312, 101031873, -3963904, 134185, -4416, 161, -8, 1

Offset: 0

Author

Peter Luschny, Jul 07 2019

Keywords

• sign
• tabl

Examples

			Array starts:
[0] 1,  0,   0,     0,      0,         0,          0,             0, ... A000007
[1] 1, -1,   5,   -61,   1385,    -50521,    2702765,    -199360981, ... A028296
[2] 1, -2,  16,  -272,   7936,   -353792,   22368256,   -1903757312, ... A000182
[3] 1, -3,  33,  -723,  25953,  -1376643,  101031873,   -9795436563, ... A326328
[4] 1, -4,  56, -1504,  64256,  -3963904,  332205056,  -36246728704, ...
[5] 1, -5,  85, -2705, 134185,  -9451805,  892060285, -108357876905, ...
[6] 1, -6, 120, -4416, 249600, -19781376, 2078100480, -278400270336, ...
       A045944,
Seen as a triangle:
[0] [1]
[1] [0, 1]
[2] [0, -1,      1]
[3] [0, 5,       -2,      1]
[4] [0, -61,     16,      -3,    1]
[5] [0, 1385,    -272,    33,    -4,    1]
[6] [0, -50521,  7936,    -723,  56,    -5, 1]
[7] [0, 2702765, -353792, 25953, -1504, 85, -6, 1]
		

Links

• Eric Weisstein's World of Mathematics, Mittag-Leffler Function

Crossrefs

Rows: A000007 (row 0), A028296 (row 1), A000182 (row 2), A326328(row 3).

Columns: A045944 (col. 2).

Cf. A326476 (m=2, p>=0), this sequence (m=2, p<=0), A326474 (m=3, p>=0), A326475 (m=3, p<=0).

Programs

Mathematica

cl[m_, p_, len_] := CoefficientList[
   Series[FunctionExpand[MittagLefflerE[m, z]^p], {z, 0, len}], z];
MLPower[m_, 0,  len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];
MLPower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;
For[n = 0, n < 8, n++, Print[MLPower[2, -n, 8]]]

Sage

def MLPower(m, p, len):
    if p == 0: return [p^k for k in (0..len-1)]
    f = [i/m for i in (1..m-1)]
    h = lambda x: hypergeometric([], f, (x/m)^m)
    g = [v for v in taylor(h(x)^p, x, 0, (len-1)*m).list() if v != 0]
    return [factorial(m*k)*v for (k, v) in enumerate(g)]
for p in (0..6): print(MLPower(2, -p, 9))