A326327 - OEIS

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.

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..54.

Eric Weisstein's World of Mathematics, Mittag-Leffler Function

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, ...

A045944,

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];

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

(Sage)

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

Adjacent sequences: A326324 A326325 A326326 * A326328 A326329 A326330

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jul 07 2019

STATUS

approved