A336179 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^3.

1, 1, 1, 1, 0, 1, 1, -1, -6, 1, 1, -2, -11, 0, 1, 1, -3, -14, 47, 90, 1, 1, -4, -15, 136, 241, 0, 1, 1, -5, -14, 261, 106, -2281, -1680, 1, 1, -6, -11, 416, -639, -8492, -3779, 0, 1, 1, -7, -6, 595, -2294, -17523, 35344, 104831, 34650, 1, 1, -8, 1, 792, -5135, -25624, 188049, 395008, -110207, 0, 1

Offset

0,9

Comments

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z - k*z*x - (k-1)*x*y*z).

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + k*x*y*z).

Links

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

Example

Square array begins:
  1,  1,     1,     1,      1,      1, ...
  1,  0,    -1,    -2,     -3,     -4, ...
  1, -6,   -11,   -14,    -15,    -14, ...
  1,  0,    47,   136,    261,    416, ...
  1, 90,   241,   106,   -639,  -2294, ...
  1,  0, -2281, -8492, -17523, -25624, ...

Mathematica

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[(-k)^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

Crossrefs

Columns k=0-3 give: A000012, A245086, A336181, A336182.

Main diagonal gives A336180.

Cf. A307884, A336163.

Sequence in context: A128423 A345628 A364129 * A350034 A133419 A010134

Adjacent sequences: A336176 A336177 A336178 * A336180 A336181 A336182

Keyword

sign,tabl

Author

Seiichi Manyama, Jul 10 2020

Status

approved