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%I #18 Jan 14 2024 09:13:10
%S 1,1,6,1,1,30,1,1,60,140,1,1,45,105,630,1,1,20,140,252,2772,1,1,6,14,
%T 1260,693,12012,1,1,900,2100,945,5940,10296,51480,1,1,3,1,945,189,
%U 1287,6435,218790,1,1,100,700,420,660,12012,780,145860,923780
%N Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.
%C The row sums of the triangle, seen in its rational form A368847(n)/ A368848(n), are the unsigned Bernoulli numbers |B(2n)|. To get the signed Bernoulli numbers B(2n), one only needs to change the sign factor in the definition of A368846 from (-1)^(n + k) to (-1)^(n + 1).
%H Paolo Xausa, <a href="/A368848/b368848.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H Thomas Curtright, <a href="https://doi.org/10.48550/arXiv.2401.00586">Scale Invariant Scattering and the Bernoulli Numbers</a>, arXiv:2401.00586 [math-ph], Jan 2024.
%e Triangle starts:
%e [0] [1]
%e [1] [1, 6]
%e [2] [1, 1, 30]
%e [3] [1, 1, 60, 140]
%e [4] [1, 1, 45, 105, 630]
%e [5] [1, 1, 20, 140, 252, 2772]
%e [6] [1, 1, 6, 14, 1260, 693, 12012]
%e [7] [1, 1, 900, 2100, 945, 5940, 10296, 51480]
%e [8] [1, 1, 3, 1, 945, 189, 1287, 6435, 218790]
%t A368846[n_,k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1,n] Binomial[2n+1, 2k]];
%t Denominator[MapIndexed[Take[#,First[#2]]&, Inverse[PadRight[Table[ A368846[n, k], {n,0,10},{k,0,n}]]]]] (* _Paolo Xausa_, Jan 08 2024 *)
%o (SageMath)
%o M = matrix(ZZ, 10, 10, lambda n, k: A368846(n, k) if k <= n else 0)
%o I = M.inverse()
%o for n in range(9): print([I[n][k].denominator() for k in range(n+1)])
%Y Cf. A368846 (inverse), A368847 (numerator), A002457 (main diagonal), A369134, A369135, A000367/A002445 (Bernoulli(2n)).
%K nonn,tabl,frac
%O 0,3
%A _Peter Luschny_, Jan 07 2024
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