A326476 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, 1, 2, 1, 0, 1, 8, 3, 1, 0, 1, 32, 21, 4, 1, 0, 1, 128, 183, 40, 5, 1, 0, 1, 512, 1641, 544, 65, 6, 1, 0, 1, 2048, 14763, 8320, 1205, 96, 7, 1, 0, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1, 0, 1, 32768, 1195743, 2099200, 628805, 64896, 3787, 176, 9, 1

Offset: 0

Author

Peter Luschny, Jul 08 2019

Keywords

• nonn
• tabl

Examples

			Array starts:
  [0] 1, 0,   0,    0,      0,        0,          0,            0, ... A000007
  [1] 1, 1,   1,    1,      1,        1,          1,            1, ... A000012
  [2] 1, 2,   8,   32,    128,      512,       2048,         8192, ... A081294
  [3] 1, 3,  21,  183,   1641,    14763,     132861,      1195743, ... A054879
  [4] 1, 4,  40,  544,   8320,   131584,    2099200,     33562624, ... A092812
  [5] 1, 5,  65, 1205,  26465,   628805,   15424865,    382964405, ... A121822
  [6] 1, 6,  96, 2256,  64896,  2086656,   71172096,   2499219456, ...
  [7] 1, 7, 133, 3787, 134953,  5501167,  243147373,  11266376947, ...
  [8] 1, 8, 176, 5888, 250496, 12397568,  676591616,  39316226048, ...
  [9] 1, 9, 225, 8649, 427905, 24943689, 1624354785, 114066126729, ...
        A000567,
Seen as a triangle:
  1;
  0, 1;
  0, 1,    1;
  0, 1,    2,      1;
  0, 1,    8,      3,      1;
  0, 1,   32,     21,      4,     1;
  0, 1,  128,    183,     40,     5,    1;
  0, 1,  512,   1641,    544,    65,    6,   1;
  0, 1, 2048,  14763,   8320,  1205,   96,   7, 1;
  0, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1;
		

Crossrefs

Rows n=0..5 give A000007, A000012, A081294, A054879, A092812, A121822.

Columns include: A000567.

Main diagonal gives A381459.

Variant: A286899.

Cf. A326474 (m=3, p>=0), A326475 (m=3, p<=0), A326327 (m=2, p<=0), this sequence (m=2, p>=0).

Programs

Mathematica

(* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[2, n, 8]]]

PARI

a(n, k) = (2*k)!*polcoef(cosh(x+x*O(x^(2*k)))^n, 2*k); \\ Seiichi Manyama, May 11 2025

Sage

# uses[MLPower from A326327]
for n in (0..6): print(MLPower(2, n, 9))

Formula

A(n,k) = (2*k)! * [x^(2*k)] cosh(x)^n. - Seiichi Manyama, May 11 2025