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Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.
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%I #24 Dec 08 2023 11:38:50

%S 1,1,-1,1,0,-1,1,1,-1,-1,1,1,0,-1,-1,1,3,1,-1,-3,-1,1,2,1,0,-1,-2,-1,

%T 1,5,3,1,-1,-3,-5,-1,1,3,1,1,0,-1,-1,-3,-1,1,7,5,1,1,-1,-1,-5,-7,-1,1,

%U 4,3,2,1,0,-1,-2,-3,-4,-1,1,0,7,5,3,1,-1,-3,-5,-7,-9,-1

%N Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

%C This array generalizes A367727.

%H Stefano Spezia, <a href="/A367824/b367824.txt">First 151 antidiagonals of the array</a>

%F A(1, n) = -A026741(n-1) for n > 0.

%F A(2, n) = -A060819(n-2) for n > 2.

%F A(3, n) = -A060789(n-3) for n > 3.

%F A(4, n) = -A106609(n-4) for n > 3.

%F A(5, n) = -A106611(n-5) for n > 4.

%F A(6, n) = -A051724(n-6) for n > 5.

%F A(7, n) = -A106615(n-7) for n > 6.

%F A(8, n) = -A106617(n-8) = A231190(n) for n > 7.

%F A(9, n) = -A106619(n-9) for n > 8.

%F A(10, n) = -A106612(n-10) for n > 9.

%e The array of the fractions begins:

%e 1, -1, -1, -1, -1, -1, -1, -1, ...

%e 1, 0, -1/3, -1/2, -3/5, -2/3, -5/7, -3/4, ...

%e 1, 1/3, 0, -1/5, -1/3, -3/7, -1/2, -5/9, ...

%e 1, 1/2, 1/5, 0, -1/7, -1/4, -1/3, -2/5, ...

%e 1, 3/5, 1/3, 1/7, 0, -1/9, -1/5, -3/11, ...

%e 1, 2/3, 3/7, 1/4, 1/9, 0, -1/11, -1/6, ...

%e 1, 5/7, 1/2, 1/3, 1/5, 1/11, 0, -1/13, ...

%e 1, 3/4, 5/9, 2/5, 3/11, 1/6, 1/13, 0, ...

%e ...

%e The array of the numerators begins:

%e 1, -1, -1, -1, -1, -1, -1, -1, ...

%e 1, 0, -1, -1, -3, -2, -5, -3, ...

%e 1, 1, 0, -1, -1, -3, -1, -5, ...

%e 1, 1, 1, 0, -1, -1, -1, -2, ...

%e 1, 3, 1, 1, 0, -1, -1, -3, ...

%e 1, 2, 3, 1, 1, 0, -1, -1, ...

%e 1, 5, 1, 1, 1, 1, 0, -1, ...

%e 1, 3, 5, 2, 3, 1, 1, 0, ...

%e ...

%t A[0,0]=1; A[n_,k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

%Y Cf. A367825 (denominator), A367826 (antidiagonal sums).

%Y Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

%K sign,base,frac,look,tabl

%O 0,17

%A _Stefano Spezia_, Dec 02 2023