A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

1, 1, 2, 1, 3, 6, 1, 4, 3, 26, 1, 5, 20, 33, 150, 1, 6, 15, 44, 15, 1082, 1, 7, 42, 115, 380, 273, 9366, 1, 8, 7, 366, 285, 4108, 1491, 94586, 1, 9, 72, 91, 4074, 3535, 17780, 38001, 1091670, 1, 10, 45, 776, 70, 11334, 26355, 269348, 17295, 14174522, 1, 11, 110, 531, 10440, 2149, 189714, 458555, 4663060, 566733, 204495126

Offset: 0

Author

Mohammed Yaseen, Jul 22 2024

Keywords

• nonn
• tabl
• frac

Examples

			Array begins:
+-----+--------------------------------------------------------------+
| n\k |       2     3       4       5        6      7         8  ... |
+-----+--------------------------------------------------------------+
|  0  |       1     1       1       1        1      1         1  ... |
|  1  |       2     3       4       5        6      7         8  ... |
|  2  |       6     3      20      15       42      7        72  ... |
|  3  |      26    33      44     115      366     91       776  ... |
|  4  |     150    15     380     285     4074     70     10440  ... |
|  5  |    1082   273    4108    3535    11334   2149    174728  ... |
|  6  |    9366  1491   17780   26355   189714   3311   3525192  ... |
|  7  |   94586 38001  269348  458555  3706518 285929  11870648  ... |
|  8  | 1091670 17295 4663060 1139685 82749954 220430 319735800  ... |
| ... |     ...   ...     ...     ...      ...    ...       ...  ... |
+-----+--------------------------------------------------------------+
		

Crossrefs

Cf. A374896 (denominators).

Cf. A000629, A008292, A121376, A276805.

Programs

PARI

T(n,k) = numerator(polylog(-n, 1/k));
matrix(7,7,n,k,T(n-1, k+1)) \\ Michel Marcus, Aug 04 2024

Formula

T(n,k) = numerator(polylog(-n, 1/k)).

T(n,k) = numerator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).

T(0,k) = 1.

T(1,k) = k.

T(2,k) = A276805(k-1).

T(n,2) = A000629(n).

T(n,n) = A121376(n).