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
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Programs
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PARI
a(n, k) = prod(j=0, n-1, n-k*j);
Formula
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.