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A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.

%I A374896 #6 Aug 19 2024 11:51:59
%S A374896 1,2,1,3,4,1,4,9,2,1,5,16,27,8,1,6,25,32,27,1,1,7,36,125,128,81,4,1,8,
%T A374896 49,27,625,128,243,4,1,9,64,343,216,3125,512,243,16,1,10,81,256,2401,
%U A374896 81,3125,1024,729,1,1,11,100,729,2048,16807,972,15625,4096,2187,4,1
%N A374896 Array read by falling antidiagonals: T(n,k) = denominator(Sum_{x>0} (x^n)/(k^x)); n >= 0 and k >= 2.
%F A374896 T(n,k) = denominator(polylog(-n, 1/k)).
%F A374896 T(n,k) = denominator(1/(k-1)^(n+1) * Sum_{m=1..n} A008292(n,m)*k^m).
%F A374896 T(0,k) = k-1.
%F A374896 T(1,k) = (k-1)^2.
%F A374896 T(2,k) = A277542(k-1).
%F A374896 T(n,2) = 1.
%F A374896 T(n,n) = A121985(n).
%e A374896 Array begins:
%e A374896 +-----+-----------------------------------------------+
%e A374896 | n\k |   2    3     4    5      6     7       8  ... |
%e A374896 +-----+-----------------------------------------------+
%e A374896 |  0  |   1    2     3    4      5     6       7  ... |
%e A374896 |  1  |   1    4     9   16     25    36      49  ... |
%e A374896 |  2  |   1    2    27   32    125    27     343  ... |
%e A374896 |  3  |   1    8    27  128    625   216    2401  ... |
%e A374896 |  4  |   1    1    81  128   3125    81   16807  ... |
%e A374896 |  5  |   1    4   243  512   3125   972  117649  ... |
%e A374896 |  6  |   1    4   243 1024  15625   486  823543  ... |
%e A374896 |  7  |   1   16   729 4096  78125 11664  823543  ... |
%e A374896 |  8  |   1    1  2187 2048 390625  2187 5764801  ... |
%e A374896 | ... | ...  ...   ...  ...    ...   ...     ...  ... |
%e A374896 +-----+-----------------------------------------------+
%o A374896 (PARI) T(n,k) = denominator(polylog(-n, 1/k));
%o A374896 matrix(7,7,n, k, T(n-1,k+1)) \\ _Michel Marcus_, Aug 04 2024
%Y A374896 Cf. A374895 (numerators).
%Y A374896 Cf. A008292, A121985, A277542.
%K A374896 nonn,tabl,frac
%O A374896 0,2
%A A374896 _Mohammed Yaseen_, Aug 03 2024