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A368847 Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.

%I A368847 #17 Jan 14 2024 09:12:30
%S A368847 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 A368847 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 A368847 43867,43867,750167,6583,943,1129,217,2,1,0,0,1222277,174611,627073,1540967,28399,53,47,23,1,1
%N A368847 Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.
%C A368847 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 A368847 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 A368847 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 A368847 Peter Luschny, <a href="https://github.com/PeterLuschny/Gists/blob/main/ThreeGracesDanceTheBernoulliWaltz.ipynb">Illustrating the polynomials</a>.
%e A368847 [0] [1]
%e A368847 [1] [0, 1]
%e A368847 [2] [0, 0,    1]
%e A368847 [3] [0, 0,    1,     1]
%e A368847 [4] [0, 0,    1,     1,    1]
%e A368847 [5] [0, 0,    1,     3,    1,   1]
%e A368847 [6] [0, 0,    1,     1,   17,   1,   1]
%e A368847 [7] [0, 0,  691,   691,   59,  41,   5, 1]
%e A368847 [8] [0, 0,   14,     2,  359,   8,   4, 1, 1]
%e A368847 [9] [0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1]
%t A368847 A368846[n_,k_]:=If[k==0,Boole[n==0],(-1)^(n+k)2Binomial[2k-1,n]Binomial[2n+1,2k]];
%t A368847 Numerator[MapIndexed[Take[#,First[#2]]&,Inverse[PadRight[Table[A368846[n,k],{n,0,10},{k,0,n}]]]]] (* _Paolo Xausa_, Jan 08 2024 *)
%o A368847 (SageMath)
%o A368847 M = matrix(ZZ, 10, 10, lambda n, k: A368846(n, k) if k <= n else 0)
%o A368847 I = M.inverse()
%o A368847 for n in range(9): print([I[n][k].numerator() for k in range(n+1)])
%Y A368847 Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).
%K A368847 nonn,frac,tabl
%O A368847 0,19
%A A368847 _Peter Luschny_, Jan 07 2024