A394320 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/(cos(x) + sin(x)).

1, 1, 0, 1, 3, 0, 1, 8, 57, 0, 1, 15, 256, 2763, 0, 1, 24, 705, 17408, 250737, 0, 1, 35, 1536, 63375, 2031616, 36581523, 0, 1, 48, 2905, 175104, 9208065, 362283008, 7828053417, 0, 1, 63, 4992, 407435, 30867456, 1966158735, 91620376576, 2309644635483, 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) = 2*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, ...
  0,      3,       8,      15,       24, ...
  0,     57,     256,     705,     1536, ...
  0,   2763,   17408,   63375,   175104, ...
  0, 250737, 2031616, 9208065, 30867456, ...

Mathematica

A394320[n_, k_] := A394320[n, k] = If[n == 0, 1, 2*k*(k+1)*A394320[n-1, k+2] - k^2*A394320[n-1, k]];
Table[A394320[k, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2026 *)

Prog

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

Crossrefs

Columns k=0..2 give A000007, A000281, A253165.

Cf. A341268, A395232, A395233.

Sequence in context: A385910 A378062 A174860 * A157391 A099097 A152150

Adjacent sequences: A394317 A394318 A394319 * A394321 A394322 A394323

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 16 2026

Status

approved