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 A368848 Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846. 4
 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, 1260, 693, 12012, 1, 1, 900, 2100, 945, 5940, 10296, 51480, 1, 1, 3, 1, 945, 189, 1287, 6435, 218790, 1, 1, 100, 700, 420, 660, 12012, 780, 145860, 923780 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened). Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024. EXAMPLE Triangle starts: [0] [1] [1] [1, 6] [2] [1, 1, 30] [3] [1, 1, 60, 140] [4] [1, 1, 45, 105, 630] [5] [1, 1, 20, 140, 252, 2772] [6] [1, 1, 6, 14, 1260, 693, 12012] [7] [1, 1, 900, 2100, 945, 5940, 10296, 51480] [8] [1, 1, 3, 1, 945, 189, 1287, 6435, 218790] MATHEMATICA A368846[n_, k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]]; Denominator[MapIndexed[Take[#, First[#2]]&, Inverse[PadRight[Table[ A368846[n, k], {n, 0, 10}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 08 2024 *) PROG (SageMath) M = matrix(ZZ, 10, 10, lambda n, k: A368846(n, k) if k <= n else 0) I = M.inverse() for n in range(9): print([I[n][k].denominator() for k in range(n+1)]) CROSSREFS Cf. A368846 (inverse), A368847 (numerator), A002457 (main diagonal), A369134, A369135, A000367/A002445 (Bernoulli(2n)). Sequence in context: A296548 A201461 A340475 * A265603 A174186 A111578 Adjacent sequences: A368845 A368846 A368847 * A368849 A368850 A368851 KEYWORD nonn,tabl,frac AUTHOR Peter Luschny, Jan 07 2024 STATUS approved

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Last modified June 23 01:45 EDT 2024. Contains 373629 sequences. (Running on oeis4.)