OFFSET
0,8
LINKS
Eric Weisstein's World of Mathematics, Mittag-Leffler Function
FORMULA
A(n, k) = (2*k)! [x^k] MittagLefflerE(2, x)^(-n).
EXAMPLE
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, ...
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]
MATHEMATICA
cl[m_, p_, len_] := CoefficientList[
Series[FunctionExpand[MittagLefflerE[m, z]^p], {z, 0, len}], z];
omegapower[m_, 0, len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];
omegapower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;
For[n = 0, n < 8, n++, Print[omegapower[2, n, 8]]]
PROG
(Sage)
def OmegaPower(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 OmegaPower(2, -p, 9)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jul 07 2019
STATUS
editing