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