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A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (k*n+j)*x).

%I A382347 #20 May 18 2025 09:58:24
%S A382347 1,1,1,1,3,2,1,5,26,6,1,7,74,342,24,1,9,146,1650,5944,120,1,11,242,
%T A382347 4578,48504,127860,720,1,13,362,9774,189144,1763100,3272688,5040,1,15,
%U A382347 506,17886,520024,9660840,76223664,97053936,40320,1,17,674,29562,1164024,34201080,586813968,3817038960,3270729600,362880
%N A382347 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n] Product_{j=0..n} (1 + (k*n+j)*x).
%F A382347 A(n,k) = A165675((k+1)*n,k*n).
%F A382347 A(n,k) = Sum_{j=0..n} (j+1) * (k*n)^j * |Stirling1(n+1,j+1)|.
%F A382347 A(n,k) = (n+1)! * Sum_{j=0..n} (-1)^j * binomial(-k*n,j)/(n+1-j).
%F A382347 A(n,k) = ((k+1)*n)!/(k*n)! * (1 + k*n * Sum_{j=1..n} 1/(k*n+j)).
%e A382347 Square array begins:
%e A382347     1,      1,       1,       1,        1,        1, ...
%e A382347     1,      3,       5,       7,        9,       11, ...
%e A382347     2,     26,      74,     146,      242,      362, ...
%e A382347     6,    342,    1650,    4578,     9774,    17886, ...
%e A382347    24,   5944,   48504,  189144,   520024,  1164024, ...
%e A382347   120, 127860, 1763100, 9660840, 34201080, 93638820, ...
%o A382347 (PARI) a(n, k) = sum(j=0, n, (j+1)*(k*n)^j*abs(stirling(n+1, j+1, 1)));
%Y A382347 Columns k=0..3 give A000142, A384024, A383678, A382349.
%Y A382347 Main diagonal gives A380707.
%Y A382347 Cf. A165675.
%K A382347 nonn,tabl
%O A382347 0,5
%A A382347 _Seiichi Manyama_, May 18 2025