A395215 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where A(n,k) = (2*n)! * [x^(2*n)] 1/((1 - x)*(1 - 3*x))^(k/2).

1, 1, 0, 1, 9, 0, 1, 26, 681, 0, 1, 51, 2904, 148905, 0, 1, 84, 7965, 786960, 64805265, 0, 1, 125, 17544, 2652615, 396789120, 46896669225, 0, 1, 174, 33705, 7087680, 1547328825, 321413702400, 50831084252025, 0, 1, 231, 58896, 16296525, 4761832320, 1402608836475, 381841394457600, 77061447551313225, 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+2) * A(n-1,k+4) + 3*k*(k+1) * A(n-1,k+2) for n > 0.

A(n,k) = (2*n)! * Sum_{j=0..n} 16^j * (-3)^(n-j) * binomial(n+j+k/2-1,n+j) * binomial(n+j,2*j).

Example

Square array begins:
  1,        1,         1,          1,          1, ...
  0,        9,        26,         51,         84, ...
  0,      681,      2904,       7965,      17544, ...
  0,   148905,    786960,    2652615,    7087680, ...
  0, 64805265, 396789120, 1547328825, 4761832320, ...

Mathematica

A395215[n_, k_] := A395215[n, k] = If[n == 0, 1, k*(k+2)*A395215[n-1, k+4] + 3*k*(k+1)*A395215[n-1, k+2]];
Table[A395215[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+2)*a(n-1, k+4)+3*k*(k+1)*a(n-1, k+2));

Crossrefs

Columns k=0..2 give A000007, A395216, (2*n)! * A096053(n).

Cf. A395148, A395201.

Sequence in context: A065471 A310000 A199284 * A189186 A221429 A221507

Adjacent sequences: A395212 A395213 A395214 * A395216 A395217 A395218

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 16 2026

Status

approved