A368848 Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.

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

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