A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).

1, 1, 2, 1, 2, 6, 1, 2, 4, 20, 1, 2, 2, 8, 70, 1, 2, 0, -2, 16, 252, 1, 2, -2, -10, -14, 32, 924, 1, 2, -4, -16, -22, -32, 64, 3432, 1, 2, -6, -20, -10, 12, -30, 128, 12870, 1, 2, -8, -22, 20, 118, 174, 64, 256, 48620, 1, 2, -10, -22, 66, 242, 304, 344, 346, 512, 184756

Offset

0,3

Links

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

Formula

G.f. of column k: 1/sqrt(1 - 4*x*(1-x)^k).

n*T(n,k) = 2 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

Example

Square array begins:
    1,  1,   1,   1,   1,   1,    1, ...
    2,  2,   2,   2,   2,   2,    2, ...
    6,  4,   2,   0,  -2,  -4,   -6, ...
   20,  8,  -2, -10, -16, -20,  -22, ...
   70, 16, -14, -22, -10,  20,   66, ...
  252, 32, -32,  12, 118, 242,  342, ...
  924, 64, -30, 174, 304,  82, -678, ...

Prog

(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(2*j, j)*binomial(k*j, n-j));

Crossrefs

Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.

Main diagonal gives A361835.

Cf. A361830.

Sequence in context: A083773 A129116 A096179 * A166350 A357124 A210227

Adjacent sequences: A361831 A361832 A361833 * A361835 A361836 A361837

Keyword

sign,tabl

Author

Seiichi Manyama, Mar 26 2023

Status

approved