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

Author

Peter Luschny, Jan 07 2024

Keywords

• nonn
• tabl
• frac

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).

Examples

			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]
		

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.

Crossrefs

Cf. A368846 (inverse), A368847 (numerator), A002457 (main diagonal), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

Programs

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 *)

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)])