A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)*2*binomial(2*k - 1, n)* binomial(2*n + 1, 2*k) for k > 0, and k^n for k = 0.

1, 0, 6, 0, 0, 30, 0, 0, -70, 140, 0, 0, 0, -840, 630, 0, 0, 0, 924, -6930, 2772, 0, 0, 0, 0, 18018, -48048, 12012, 0, 0, 0, 0, -12870, 216216, -300300, 51480, 0, 0, 0, 0, 0, -350064, 2042040, -1750320, 218790, 0, 0, 0, 0, 0, 184756, -5542680, 16628040, -9699690, 923780

Offset: 0

Author

Peter Luschny, Jan 07 2024

Keywords

• sign
• tabl

Comments

The row sums of the inverse triangle (A368847/A368848) 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 from (-1)^(n + k) to (-1)^(n + 1).

Conjecture: |Sum_{j=0..k} T(k + j, k)| = A229580(k + 1) for k >= 0.

Examples

			[0] [1]
[1] [0, 6]
[2] [0, 0,  30]
[3] [0, 0, -70,  140]
[4] [0, 0,   0, -840,    630]
[5] [0, 0,   0,  924,  -6930,   2772]
[6] [0, 0,   0,    0,  18018,  -48048,   12012]
[7] [0, 0,   0,    0, -12870,  216216, -300300,    51480]
[8] [0, 0,   0,    0,      0, -350064, 2042040, -1750320, 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.
• Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025.

Crossrefs

Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.

Programs

Mathematica

A368846[n_,k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];
Table[A368846[n, k], {n,0,10}, {k,0,n}] (* Paolo Xausa, Jan 08 2024 *)

SageMath

def A368846(n, k):
    if k == 0: return k^n
    if k  > n: return 0
    return (-1)^(n + k)*2*binomial(2*k - 1, n)*binomial(2*n + 1, 2*k)
for n in range(10): print([A368846(n, k) for k in range(n+1)])