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