A336163 -id:A336163 - cp's OEIS frontend

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

Seiichi Manyama, Jul 10 2020

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).

			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, ...

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

Columns k=0-3 give: A000012, A245086, A336181, A336182.
Main diagonal gives A336180.
Cf. A307884, A336163.

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 *)

A336187 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)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 34, 63, 16, 1, 1, 6, 81, 352, 321, 32, 1, 1, 7, 186, 1685, 3946, 1683, 64, 1, 1, 8, 421, 7416, 38401, 46744, 8989, 128, 1, 1, 9, 946, 30835, 328146, 963525, 573616, 48639, 256, 1, 1, 10, 2113, 122816, 2590225, 16971876, 25346385, 7217536, 265729, 512, 1

Offset: 0

Seiichi Manyama, Jul 11 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - k * Product_{j=1..k} x_j) for k>0.

			Square array begins:
  1,  1,    1,     1,      1,        1, ...
  1,  2,    3,     4,      5,        6, ...
  1,  4,   13,    34,     81,      186, ...
  1,  8,   63,   352,   1685,     7416, ...
  1, 16,  321,  3946,  38401,   328146, ...
  1, 32, 1683, 46744, 963525, 16971876, ...

Columns k=0-3 give: A000012, A000079, A001850, A206180.
Main diagonal gives A336188.
Cf. A307883, A336163.

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

cp's OEIS Frontend

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.

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