A395200 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = (2*n)! * [x^(2*n)] (f(x)^k + f(-x)^k)/2 and f(x) = 1/(cosh(x) - sqrt(2) * sinh(x)).

1, 1, 0, 1, 3, 0, 1, 10, 45, 0, 1, 21, 256, 1743, 0, 1, 36, 849, 13840, 125625, 0, 1, 55, 2136, 61941, 1282816, 14554683, 0, 1, 78, 4525, 204576, 7214529, 181649920, 2473184805, 0, 1, 105, 8520, 554755, 29419776, 1229315061, 36478744576, 579439207623, 0

Offset

0,5

Links

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Formula

A(0,k) = 1 and A(n,k) = k*(k+1) * A(n-1,k+2) + k^2 * A(n-1,k) for n > 0.

Example

Square array begins:
  1,        1,         1,          1,          1,           1, ...
  0,        3,        10,         21,         36,          55, ...
  0,       45,       256,        849,       2136,        4525, ...
  0,     1743,     13840,      61941,     204576,      554755, ...
  0,   125625,   1282816,    7214529,   29419776,    97032025, ...
  0, 14554683, 181649920, 1229315061, 5958690816, 23118264655, ...

Mathematica

A395200[n_, k_] := A395200[n, k] = If[n == 0, 1, k*(k+1)*A395200[n-1, k+2] + k^2*A395200[n-1, k]];
Table[A395200[k, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 16 2026 *)

Prog

(PARI) a(n, k) = if(n==0, 1, k*(k+1)*a(n-1, k+2)+k^2*a(n-1, k));

Crossrefs

Columns k=0..3 give A000007, |A012494(n)|, |A024293(n+1)|, A395202.

Cf. A341268, A395201.

Sequence in context: A269939 A239731 A327027 * A145881 A232223 A380191

Adjacent sequences: A395197 A395198 A395199 * A395201 A395202 A395203

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 15 2026

Status

approved