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

%I A384216 #21 May 24 2025 05:29:37
%S A384216 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,
%T A384216 1,1,-8,42,0,-280,0,5040,1,1,-10,81,-264,-1155,0,-1575,40320,1,1,-12,
%U A384216 132,-896,0,20160,24640,0,362880,1,1,-14,195,-2040,8645,57456,-208845,-291200,99225,3628800
%N A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).
%H A384216 Seiichi Manyama, <a href="/A384216/b384216.txt">Antidiagonals n = 0..139, flattened</a>
%F A384216 A(n,k) = Product_{j=0..n-1} (n-k*j).
%F A384216 A(n,k) = k^n * FallingFactorial(n/k,n).
%F A384216 A(k*n,k) = 0 for n > 0 and k > 1.
%e A384216 Square array begins:
%e A384216     1,  1,    1,     1,     1,    1,       1, ...
%e A384216     1,  1,    1,     1,     1,    1,       1, ...
%e A384216     2,  0,   -2,    -4,    -6,   -8,     -10, ...
%e A384216     6, -3,    0,    15,    42,   81,     132, ...
%e A384216    24,  0,   40,     0,  -264, -896,   -2040, ...
%e A384216   120, 45, -280, -1155,     0, 8645,   33120, ...
%e A384216   720,  0,    0, 20160, 57456,    0, -459360, ...
%o A384216 (PARI) a(n, k) = prod(j=0, n-1, n-k*j);
%Y A384216 Columns k=3..5 give A282627(n+1)/2, A383996, A383997.
%Y A384216 Cf. A303489.
%K A384216 sign,tabl,easy
%O A384216 0,6
%A A384216 _Seiichi Manyama_, May 22 2025