proposed

approved

editing

proposed

for p in (0..6): print (MLPower(2, -p, 9))

approved

editing

editing

approved

Cf. A286899A326476 (m=2, p>=0), this sequence (m=2, p<=0), A326474 (m=3, p>=0), A326475 (m=3, p<=0).

For[n = 0, n < 8, n++, Print[MLPower[2, -n, 8]]]

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

A(n, k) = (2*k)! [x^k] MittagLefflerE(2, x)^(-n).

omegapowerMLPower[m_, 0, len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];

omegapowerMLPower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;

For[n = 0, n < 8, n++, Print[omegapowerMLPower[2, n, 8]]]

def OmegaPowerMLPower(m, p, len):

for p in (0..6): print OmegaPowerMLPower(2, -p, 9)

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

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mittag-LefflerFunction.html">Mittag-Leffler Function</a>

A(n, k) = (2*k)! [x^k] MittagLefflerE(2, x)^(-n).

Square array read by descending antidiagonals, A(n, k) = [x^k] OmegaPower(m, -n), for m = 2, n >= 0, k >= 0.

[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,

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

Columns: A045944 (col. 2).