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%I #17 Jan 14 2024 09:12:30
%S 1,0,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,3,1,1,0,0,1,1,17,1,1,0,0,691,691,
%T 59,41,5,1,0,0,14,2,359,8,4,1,1,0,0,3617,10851,1237,217,293,1,7,1,0,0,
%U 43867,43867,750167,6583,943,1129,217,2,1,0,0,1222277,174611,627073,1540967,28399,53,47,23,1,1
%N Triangle read by rows: T(n, k) = numerator(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="/A368847/b368847.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.
%H Peter Luschny, <a href="https://github.com/PeterLuschny/Gists/blob/main/ThreeGracesDanceTheBernoulliWaltz.ipynb">Illustrating the polynomials</a>.
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 0, 1]
%e [3] [0, 0, 1, 1]
%e [4] [0, 0, 1, 1, 1]
%e [5] [0, 0, 1, 3, 1, 1]
%e [6] [0, 0, 1, 1, 17, 1, 1]
%e [7] [0, 0, 691, 691, 59, 41, 5, 1]
%e [8] [0, 0, 14, 2, 359, 8, 4, 1, 1]
%e [9] [0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1]
%t A368846[n_,k_]:=If[k==0,Boole[n==0],(-1)^(n+k)2Binomial[2k-1,n]Binomial[2n+1,2k]];
%t Numerator[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].numerator() for k in range(n+1)])
%Y Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).
%K nonn,frac,tabl
%O 0,19
%A _Peter Luschny_, Jan 07 2024