A368847 Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.

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

Author

Peter Luschny, Jan 07 2024

Keywords

• nonn
• frac
• tabl

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

			[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]
		

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.
• Peter Luschny, Illustrating the polynomials.

Crossrefs

Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

Programs

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 *)

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)])