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A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

%I A303489 #4 Apr 24 2018 19:12:24
%S A303489 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 A303489 3640,45045,332640,1,1,16,312,6144,104720,1290240,8648640,1,1,18,405,
%U A303489 9576,208845,3674160,43648605,259459200,1,1,20,510,14080,375000,8648640,152152000,1703116800,8821612800
%N A303489 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 A303489 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A303489 A(n,k) = Product_{j=0..n-1} (k*j + n).
%e A303489 Square array begins:
%e A303489       1,      1,       1,       1,       1,       1,  ...
%e A303489       1,      1,       1,       1,       1,       1,  ...
%e A303489       6,      8,      10,      12,      14,      16,  ...
%e A303489      60,    105,     162,     231,     312,     405,  ...
%e A303489     840,   1920,    3640,    6144,    9576,   14080,  ...
%e A303489   15120,  45045,  104720,  208845,  375000,  623645,  ...
%e A303489 =========================================================
%e A303489 A(1,1) = 1;
%e A303489 A(2,1) = 2*3 = 6;
%e A303489 A(3,1) = 3*4*5 = 60;
%e A303489 A(4,1) = 4*5*6*7 = 840;
%e A303489 A(5,1) = 5*6*7*8*9 = 15120, etc.
%e A303489 ...
%e A303489 A(1,2) = 1;
%e A303489 A(2,2) = 2*4 = 8;
%e A303489 A(3,2) = 3*5*7 = 105;
%e A303489 A(4,2) = 4*6*8*10 = 1920;
%e A303489 A(5,2) = 5*7*9*11*13 = 45045, etc.
%e A303489 ...
%e A303489 A(1,3) = 1;
%e A303489 A(2,3) = 2*5 = 10;
%e A303489 A(3,3) = 3*6*9 = 162;
%e A303489 A(4,3) = 4*7*10*13 = 3640;
%e A303489 A(5,3) = 5*8*11*14*17 = 104720, etc.
%e A303489 ...
%t A303489 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 A303489 Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
%t A303489 Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
%Y A303489 Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
%Y A303489 Main diagonal gives A061711.
%Y A303489 Cf. A008279, A131182, A256268, A265609.
%K A303489 nonn,tabl
%O A303489 0,6
%A A303489 _Ilya Gutkovskiy_, Apr 24 2018