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URL: https://oeis.org/A326327

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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.
4
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
0,8
LINKS
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Mittag-Leffler Function
FORMULA
From Peter Bala, Feb 28 2026: (Start)
For n, k >= 0, define f_n(k, x) = (1 - x^2)*d/dx(f_n(k-1, x)) with f_n(0, x) = (1 - x^2)^(n/2). Then the row n entries of the square array equal f_n(2*k, 0), k >= 0.
E.g.f. row n: 1/cosh(x)^n.
O.g.f. for row n as a Stieljes type continued fraction: 1/(1 + n*x/(1 + 2*(n+1)*x/(1 + 3*(n+2)*x/(1 + 4*(n+3)*x/(1 + ... ))))) (see Sokal, equation 6.8). (End)
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]
MAPLE
# uses Graves' method for computing inverse function expansions
f := proc (n, k, x) option remember; if k = 0 then (1 - x^2)^(n/2) else simplify( (1 - x^2)*diff(f(n, k-1, x), x) ) end if end proc:
seq( print(seq(eval(f(n, 2*k, x), x = 0), k = 0..10)), n = 0..10); # Peter Bala, Feb 28 2026
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]]]
PROG
(SageMath)
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))
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).
Sequence in context: A334367 A367184 A370915 * A113103 A033325 A126690
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jul 07 2019
STATUS
approved