A336180 -id:A336180 - 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 *)

A354944 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-n)^(n-k).

Original entry on oeis.org

1, 0, -10, 60, 1560, -39880, -491760, 45672060, -155935360, -77656158000, 2116774828800, 166585352850620, -11925674437248000, -330617542587341880, 69148933431781898240, -543549949643024194500, -434534462104188331130880, 21521903478880966780355360

Offset: 0

Ilya Gutkovskiy, Jun 12 2022

sign

Cf. A134095, A277423, A336180, A343840, A354941, A354942, A354943.

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! (-n)^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[(-n)^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-n)^(n-k)); \\ Michel Marcus, Jun 12 2022

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} (-n)^k * x^k / k!^3.

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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