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%I #16 Mar 06 2020 04:04:51
%S 1,0,1,0,-1,1,0,5,-2,1,0,-61,16,-3,1,0,1385,-272,33,-4,1,0,-50521,
%T 7936,-723,56,-5,1,0,2702765,-353792,25953,-1504,85,-6,1,0,-199360981,
%U 22368256,-1376643,64256,-2705,120,-7,1,0,19391512145,-1903757312,101031873,-3963904,134185,-4416,161,-8,1
%N A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^(-n), for m = 2, n >= 0, k >= 0; square array read by descending antidiagonals.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mittag-LefflerFunction.html">Mittag-Leffler Function</a>
%e Array starts:
%e [0] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007
%e [1] 1, -1, 5, -61, 1385, -50521, 2702765, -199360981, ... A028296
%e [2] 1, -2, 16, -272, 7936, -353792, 22368256, -1903757312, ... A000182
%e [3] 1, -3, 33, -723, 25953, -1376643, 101031873, -9795436563, ... A326328
%e [4] 1, -4, 56, -1504, 64256, -3963904, 332205056, -36246728704, ...
%e [5] 1, -5, 85, -2705, 134185, -9451805, 892060285, -108357876905, ...
%e [6] 1, -6, 120, -4416, 249600, -19781376, 2078100480, -278400270336, ...
%e A045944,
%e Seen as a triangle:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, -1, 1]
%e [3] [0, 5, -2, 1]
%e [4] [0, -61, 16, -3, 1]
%e [5] [0, 1385, -272, 33, -4, 1]
%e [6] [0, -50521, 7936, -723, 56, -5, 1]
%e [7] [0, 2702765, -353792, 25953, -1504, 85, -6, 1]
%t cl[m_, p_, len_] := CoefficientList[
%t Series[FunctionExpand[MittagLefflerE[m, z]^p], {z, 0, len}], z];
%t MLPower[m_, 0, len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];
%t MLPower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;
%t For[n = 0, n < 8, n++, Print[MLPower[2, -n, 8]]]
%o (Sage)
%o def MLPower(m, p, len):
%o if p == 0: return [p^k for k in (0..len-1)]
%o f = [i/m for i in (1..m-1)]
%o h = lambda x: hypergeometric([], f, (x/m)^m)
%o g = [v for v in taylor(h(x)^p, x, 0, (len-1)*m).list() if v != 0]
%o return [factorial(m*k)*v for (k, v) in enumerate(g)]
%o for p in (0..6): print(MLPower(2, -p, 9))
%Y Rows: A000007 (row 0), A028296 (row 1), A000182 (row 2), A326328(row 3).
%Y Columns: A045944 (col. 2).
%Y Cf. A326476 (m=2, p>=0), this sequence (m=2, p<=0), A326474 (m=3, p>=0), A326475 (m=3, p<=0).
%K sign,tabl
%O 0,8
%A _Peter Luschny_, Jul 07 2019