A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).

1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1

Offset

0,9

Comments

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - Seiichi Manyama, Jul 11 2020

More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

T(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.

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

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

n * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-1) * T(n-2,k).

Example

Square array begins:
  1,   1,   1,    1,    1,    1,      1, ...
  1,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

Mathematica

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Crossrefs

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.

Main diagonal gives A307885.

T(n,n-1) gives A335310.

Cf. A307883, A336179.

Sequence in context: A306512 A343938 A239397 * A329221 A177858 A166967

Adjacent sequences: A307881 A307882 A307883 * A307885 A307886 A307887

Keyword

sign,tabl

Author

Seiichi Manyama, May 02 2019

Status

approved