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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).
5
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, 3640, 45045, 332640, 1, 1, 16, 312, 6144, 104720, 1290240, 8648640, 1, 1, 18, 405, 9576, 208845, 3674160, 43648605, 259459200, 1, 1, 20, 510, 14080, 375000, 8648640, 152152000, 1703116800, 8821612800
OFFSET
0,6
FORMULA
A(n,k) = Product_{j=0..n-1} (k*j + n).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
6, 8, 10, 12, 14, 16, ...
60, 105, 162, 231, 312, 405, ...
840, 1920, 3640, 6144, 9576, 14080, ...
15120, 45045, 104720, 208845, 375000, 623645, ...
=========================================================
A(1,1) = 1;
A(2,1) = 2*3 = 6;
A(3,1) = 3*4*5 = 60;
A(4,1) = 4*5*6*7 = 840;
A(5,1) = 5*6*7*8*9 = 15120, etc.
...
A(1,2) = 1;
A(2,2) = 2*4 = 8;
A(3,2) = 3*5*7 = 105;
A(4,2) = 4*6*8*10 = 1920;
A(5,2) = 5*7*9*11*13 = 45045, etc.
...
A(1,3) = 1;
A(2,3) = 2*5 = 10;
A(3,3) = 3*6*9 = 162;
A(4,3) = 4*7*10*13 = 3640;
A(5,3) = 5*8*11*14*17 = 104720, etc.
...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
CROSSREFS
Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
Main diagonal gives A061711.
Sequence in context: A171695 A179233 A141600 * A195408 A011491 A189089
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Apr 24 2018
STATUS
approved