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URL: https://oeis.org/A368847

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A368847
Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.
3
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, 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, 43867, 43867, 750167, 6583, 943, 1129, 217, 2, 1, 0, 0, 1222277, 174611, 627073, 1540967, 28399, 53, 47, 23, 1, 1
OFFSET
0,19
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
[0] [1]
[1] [0, 1]
[2] [0, 0, 1]
[3] [0, 0, 1, 1]
[4] [0, 0, 1, 1, 1]
[5] [0, 0, 1, 3, 1, 1]
[6] [0, 0, 1, 1, 17, 1, 1]
[7] [0, 0, 691, 691, 59, 41, 5, 1]
[8] [0, 0, 14, 2, 359, 8, 4, 1, 1]
[9] [0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1]
MATHEMATICA
A368846[n_, k_]:=If[k==0, Boole[n==0], (-1)^(n+k)2Binomial[2k-1, n]Binomial[2n+1, 2k]];
Numerator[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].numerator() for k in range(n+1)])
CROSSREFS
Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).
Sequence in context: A204181 A204242 A211313 * A321609 A357638 A238414
KEYWORD
nonn,frac,tabl
AUTHOR
Peter Luschny, Jan 07 2024
STATUS
approved