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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.
6
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
0,3
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.
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, 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]
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 *)
PROG
(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)])
CROSSREFS
Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.
Sequence in context: A028700 A362791 A278014 * A230337 A019157 A019184
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jan 07 2024
STATUS
approved