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

%I A374895 #10 Aug 04 2024 03:12:34
%S A374895 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,
%T A374895 273,9366,1,8,7,366,285,4108,1491,94586,1,9,72,91,4074,3535,17780,
%U A374895 38001,1091670,1,10,45,776,70,11334,26355,269348,17295,14174522,1,11,110,531,10440,2149,189714,458555,4663060,566733,204495126
%N A374895 Array read by falling antidiagonals: T(n,k) = numerator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.
%F A374895 T(n,k) = numerator(polylog(-n, 1/k)).
%F A374895 T(n,k) = numerator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
%F A374895 T(0,k) = 1.
%F A374895 T(1,k) = k.
%F A374895 T(2,k) = A276805(k-1).
%F A374895 T(n,2) = A000629(n).
%F A374895 T(n,n) = A121376(n).
%e A374895 Array begins:
%e A374895 +-----+--------------------------------------------------------------+
%e A374895 | n\k |       2     3       4       5        6      7         8  ... |
%e A374895 +-----+--------------------------------------------------------------+
%e A374895 |  0  |       1     1       1       1        1      1         1  ... |
%e A374895 |  1  |       2     3       4       5        6      7         8  ... |
%e A374895 |  2  |       6     3      20      15       42      7        72  ... |
%e A374895 |  3  |      26    33      44     115      366     91       776  ... |
%e A374895 |  4  |     150    15     380     285     4074     70     10440  ... |
%e A374895 |  5  |    1082   273    4108    3535    11334   2149    174728  ... |
%e A374895 |  6  |    9366  1491   17780   26355   189714   3311   3525192  ... |
%e A374895 |  7  |   94586 38001  269348  458555  3706518 285929  11870648  ... |
%e A374895 |  8  | 1091670 17295 4663060 1139685 82749954 220430 319735800  ... |
%e A374895 | ... |     ...   ...     ...     ...      ...    ...       ...  ... |
%e A374895 +-----+--------------------------------------------------------------+
%o A374895 (PARI) T(n,k) = numerator(polylog(-n, 1/k));
%o A374895 matrix(7,7,n,k,T(n-1, k+1)) \\ _Michel Marcus_, Aug 04 2024
%Y A374895 Cf. A374896 (denominators).
%Y A374895 Cf. A000629, A008292, A121376, A276805.
%K A374895 nonn,tabl,frac
%O A374895 0,3
%A A374895 _Mohammed Yaseen_, Jul 22 2024