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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
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.)