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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.

%I A326327 #17 Feb 16 2025 08:33:58
%S A326327 1,0,1,0,-1,1,0,5,-2,1,0,-61,16,-3,1,0,1385,-272,33,-4,1,0,-50521,
%T A326327 7936,-723,56,-5,1,0,2702765,-353792,25953,-1504,85,-6,1,0,-199360981,
%U A326327 22368256,-1376643,64256,-2705,120,-7,1,0,19391512145,-1903757312,101031873,-3963904,134185,-4416,161,-8,1
%N 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.
%H A326327 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Mittag-LefflerFunction.html">Mittag-Leffler Function</a>
%e A326327 Array starts:
%e A326327 [0] 1,  0,   0,     0,      0,         0,          0,             0, ... A000007
%e A326327 [1] 1, -1,   5,   -61,   1385,    -50521,    2702765,    -199360981, ... A028296
%e A326327 [2] 1, -2,  16,  -272,   7936,   -353792,   22368256,   -1903757312, ... A000182
%e A326327 [3] 1, -3,  33,  -723,  25953,  -1376643,  101031873,   -9795436563, ... A326328
%e A326327 [4] 1, -4,  56, -1504,  64256,  -3963904,  332205056,  -36246728704, ...
%e A326327 [5] 1, -5,  85, -2705, 134185,  -9451805,  892060285, -108357876905, ...
%e A326327 [6] 1, -6, 120, -4416, 249600, -19781376, 2078100480, -278400270336, ...
%e A326327        A045944,
%e A326327 Seen as a triangle:
%e A326327 [0] [1]
%e A326327 [1] [0, 1]
%e A326327 [2] [0, -1,      1]
%e A326327 [3] [0, 5,       -2,      1]
%e A326327 [4] [0, -61,     16,      -3,    1]
%e A326327 [5] [0, 1385,    -272,    33,    -4,    1]
%e A326327 [6] [0, -50521,  7936,    -723,  56,    -5, 1]
%e A326327 [7] [0, 2702765, -353792, 25953, -1504, 85, -6, 1]
%t A326327 cl[m_, p_, len_] := CoefficientList[
%t A326327    Series[FunctionExpand[MittagLefflerE[m, z]^p], {z, 0, len}], z];
%t A326327 MLPower[m_, 0,  len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];
%t A326327 MLPower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;
%t A326327 For[n = 0, n < 8, n++, Print[MLPower[2, -n, 8]]]
%o A326327 (Sage)
%o A326327 def MLPower(m, p, len):
%o A326327     if p == 0: return [p^k for k in (0..len-1)]
%o A326327     f = [i/m for i in (1..m-1)]
%o A326327     h = lambda x: hypergeometric([], f, (x/m)^m)
%o A326327     g = [v for v in taylor(h(x)^p, x, 0, (len-1)*m).list() if v != 0]
%o A326327     return [factorial(m*k)*v for (k, v) in enumerate(g)]
%o A326327 for p in (0..6): print(MLPower(2, -p, 9))
%Y A326327 Rows: A000007 (row 0), A028296 (row 1), A000182 (row 2), A326328(row 3).
%Y A326327 Columns: A045944 (col. 2).
%Y A326327 Cf. A326476 (m=2, p>=0), this sequence (m=2, p<=0), A326474 (m=3, p>=0), A326475 (m=3, p<=0).
%K A326327 sign,tabl
%O A326327 0,8
%A A326327 _Peter Luschny_, Jul 07 2019