A381571 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381572.

1, 1, 0, 1, 2, 0, 1, 4, 7, 0, 1, 6, 18, 38, 0, 1, 8, 33, 104, 267, 0, 1, 10, 52, 206, 735, 2232, 0, 1, 12, 75, 352, 1488, 6064, 21200, 0, 1, 14, 102, 550, 2626, 12246, 56510, 222556, 0, 1, 16, 133, 808, 4265, 21752, 112669, 581452, 2536661, 0, 1, 18, 168, 1134, 6537, 35812, 198808, 1140150, 6501267, 31010886, 0

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

Square array begins:
  1,     1,     1,      1,      1,      1,      1, ...
  0,     2,     4,      6,      8,     10,     12, ...
  0,     7,    18,     33,     52,     75,    102, ...
  0,    38,   104,    206,    352,    550,    808, ...
  0,   267,   735,   1488,   2626,   4265,   6537, ...
  0,  2232,  6064,  12246,  21752,  35812,  55944, ...
  0, 21200, 56510, 112669, 198808, 327010, 512934, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, 2*k*sum(j=0, n, binomial(2*n-j+2*k, j)/(2*n-j+2*k)*a(n-j, j)));

Crossrefs

Columns k=0..1 give A000007, A381572.

Cf. A379598, A381573.

Sequence in context: A287318 A329020 A351640 * A173003 A378236 A335461

Adjacent sequences: A381568 A381569 A381570 * A381572 A381573 A381574

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Feb 28 2025

Status

approved