A395232 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) + sqrt(2) * sin(x)).

1, 1, 0, 1, 5, 0, 1, 14, 157, 0, 1, 27, 736, 12425, 0, 1, 44, 2097, 81584, 1836697, 0, 1, 65, 4696, 308187, 15506176, 436366445, 0, 1, 90, 9085, 879584, 73033857, 4502664704, 152053957237, 0, 1, 119, 15912, 2105765, 253593856, 25415053947, 1854243260416, 73053601590065, 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) = 3*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,       5,       14,       27,        44, ...
  0,     157,      736,     2097,      4696, ...
  0,   12425,    81584,   308187,    879584, ...
  0, 1836697, 15506176, 73033857, 253593856, ...

Mathematica

A395232[n_, k_] := A395232[n, k] = If[n == 0, 1, 3*k*(k+1)*A395232[n-1, k+2] - k^2*A395232[n-1, k]];
Table[A395232[k, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2026 *)

Prog

(PARI) a(n, k) = if(n==0, 1, 3*k*(k+1)*a(n-1, k+2)-k^2*a(n-1, k));

Crossrefs

Columns k=0..2 give A000007, A156134, A156108(n)/3.

Cf. A341268, A394320, A395233, A395236.

Sequence in context: A379489 A221800 A291774 * A395201 A222061 A378981

Adjacent sequences: A395229 A395230 A395231 * A395233 A395234 A395235

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 16 2026

Status

approved