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%I #4 Apr 24 2018 19:12:24
%S 1,1,1,1,1,6,1,1,8,60,1,1,10,105,840,1,1,12,162,1920,15120,1,1,14,231,
%T 3640,45045,332640,1,1,16,312,6144,104720,1290240,8648640,1,1,18,405,
%U 9576,208845,3674160,43648605,259459200,1,1,20,510,14080,375000,8648640,152152000,1703116800,8821612800
%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A(n,k) = Product_{j=0..n-1} (k*j + n).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 6, 8, 10, 12, 14, 16, ...
%e 60, 105, 162, 231, 312, 405, ...
%e 840, 1920, 3640, 6144, 9576, 14080, ...
%e 15120, 45045, 104720, 208845, 375000, 623645, ...
%e =========================================================
%e A(1,1) = 1;
%e A(2,1) = 2*3 = 6;
%e A(3,1) = 3*4*5 = 60;
%e A(4,1) = 4*5*6*7 = 840;
%e A(5,1) = 5*6*7*8*9 = 15120, etc.
%e ...
%e A(1,2) = 1;
%e A(2,2) = 2*4 = 8;
%e A(3,2) = 3*5*7 = 105;
%e A(4,2) = 4*6*8*10 = 1920;
%e A(5,2) = 5*7*9*11*13 = 45045, etc.
%e ...
%e A(1,3) = 1;
%e A(2,3) = 2*5 = 10;
%e A(3,3) = 3*6*9 = 162;
%e A(4,3) = 4*7*10*13 = 3640;
%e A(5,3) = 5*8*11*14*17 = 104720, etc.
%e ...
%t Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
%t Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
%t Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
%Y Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
%Y Main diagonal gives A061711.
%Y Cf. A008279, A131182, A256268, A265609.
%K nonn,tabl
%O 0,6
%A _Ilya Gutkovskiy_, Apr 24 2018
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