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).

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

Links

Table of n, a(n) for n=0..54.

Index entries for sequences related to factorial numbers

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.

Cf. A008279, A131182, A256268, A265609.

Sequence in context: A171695 A179233 A141600 * A195408 A011491 A189089

Adjacent sequences: A303486 A303487 A303488 * A303490 A303491 A303492

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Apr 24 2018

Status

approved