A383997 -id:A383997 - 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.

A384242 a(n) = Product_{k=0..n-1} (4n-5k).

Original entry on oeis.org

1, 4, 24, 168, 1056, 0, -229824, -7233408, -162860544, -2573835264, 0, 2333140153344, 131053381595136, 4948323499671552, 124773727026364416, 0, -256422032696998232064, -20710128948965418074112, -1096668276542495972130816, -37948699305215165278715904, 0

Offset: 0

Seiichi Manyama, May 22 2025

Cf. A061924, A343446.

Cf. A383997.

PARI
```
a(n) = prod(k=0, n-1, 4*n-5*k);
```

def a(n): return 5^n*falling_factorial(4*n/5, n)

a(n) = 5^n * FallingFactorial(4*n/5,n).
a(n) = n! * [x^n] (1 + 5*x)^(4*n/5).
a(n) = 4 * (-1)^(n-1) * A383997(n) for n > 0.
a(5*n) = 0 for n > 0.
D-finite with recurrence a(n) +8*n*(4*n-15)*(4*n-5)*(n-5)*(2*n-5)*a(n-5)=0. - R. J. Mathar, May 26 2025

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).

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PARI

Formula

A384242 a(n) = Product_{k=0..n-1} (4n-5k).

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Author

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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

A384242 a(n) = Product_{k=0..n-1} (4*n-5*k).

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Author

Keywords

Crossrefs

Programs

PARI

Sage

Formula

A384242 a(n) = Product_{k=0..n-1} (4n-5k).