A383996 -id:A383996 - cp's OEIS frontend

A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, 0, 0, 120, 1, 1, -6, 15, 40, 45, 720, 1, 1, -8, 42, 0, -280, 0, 5040, 1, 1, -10, 81, -264, -1155, 0, -1575, 40320, 1, 1, -12, 132, -896, 0, 20160, 24640, 0, 362880, 1, 1, -14, 195, -2040, 8645, 57456, -208845, -291200, 99225, 3628800

Offset: 0

Seiichi Manyama, May 22 2025

			Square array begins:
    1,  1,    1,     1,     1,    1,       1, ...
    1,  1,    1,     1,     1,    1,       1, ...
    2,  0,   -2,    -4,    -6,   -8,     -10, ...
    6, -3,    0,    15,    42,   81,     132, ...
   24,  0,   40,     0,  -264, -896,   -2040, ...
  120, 45, -280, -1155,     0, 8645,   33120, ...
  720,  0,    0, 20160, 57456,    0, -459360, ...

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

Columns k=3..5 give A282627(n+1)/2, A383996, A383997.

Cf. A303489.

PARI
```
a(n, k) = prod(j=0, n-1, n-k*j);
```

A(n,k) = Product_{j=0..n-1} (n-k*j).
A(n,k) = k^n * FallingFactorial(n/k,n).
A(k*n,k) = 0 for n > 0 and k > 1.

A384241 a(n) = Product_{k=0..n-1} (3n-4k).

Original entry on oeis.org

1, 3, 12, 45, 0, -3465, -60480, -626535, 0, 204417675, 6227020800, 104928949125, 0, -77849405258625, -3379030566912000, -78792721832199375, 0, 104312208642352585875, 5875458349746585600000, 174954117301479619228125, 0, -362526128354588965187045625, -25100240092118201519308800000

Offset: 0

Seiichi Manyama, May 22 2025

Cf. A064352, A343445.

Cf. A383996, A384166.

PARI
```
a(n) = prod(k=0, n-1, 3*n-4*k);
```

def a(n): return 4^n*falling_factorial(3*n/4, n)

a(n) = 4^n * FallingFactorial(3*n/4,n).
a(n) = n! * [x^n] (1 + 4*x)^(3*n/4).
a(n) = 3 * (-1)^(n-1) * A383996(n) for n > 0.
a(4*n) = 0 for n > 0.

cp's OEIS Frontend

A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A384241 a(n) = Product_{k=0..n-1} (3n-4k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Sage

Formula

cp's OEIS Frontend

A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A384241 a(n) = Product_{k=0..n-1} (3*n-4*k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Sage

Formula

A384241 a(n) = Product_{k=0..n-1} (3n-4k).