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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.

%I A368846 #26 Feb 17 2025 02:52:42
%S A368846 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,
%T A368846 18018,-48048,12012,0,0,0,0,-12870,216216,-300300,51480,0,0,0,0,0,
%U A368846 -350064,2042040,-1750320,218790,0,0,0,0,0,184756,-5542680,16628040,-9699690,923780
%N 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.
%C A368846 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).
%C A368846 Conjecture: |Sum_{j=0..k} T(k + j, k)| = A229580(k + 1) for k >= 0.
%H A368846 Paolo Xausa, <a href="/A368846/b368846.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H A368846 Thomas Curtright, <a href="https://doi.org/10.48550/arXiv.2401.00586">Scale Invariant Scattering and the Bernoulli Numbers</a>, arXiv:2401.00586 [math-ph], Jan 2024.
%H A368846 Thomas L. Curtright, <a href="https://arxiv.org/abs/2502.09633">Bernoulli Partitions</a>, arXiv:2502.09633 [math.CO], 2025.
%e A368846 [0] [1]
%e A368846 [1] [0, 6]
%e A368846 [2] [0, 0,  30]
%e A368846 [3] [0, 0, -70,  140]
%e A368846 [4] [0, 0,   0, -840,    630]
%e A368846 [5] [0, 0,   0,  924,  -6930,   2772]
%e A368846 [6] [0, 0,   0,    0,  18018,  -48048,   12012]
%e A368846 [7] [0, 0,   0,    0, -12870,  216216, -300300,    51480]
%e A368846 [8] [0, 0,   0,    0,      0, -350064, 2042040, -1750320, 218790]
%t A368846 A368846[n_,k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];
%t A368846 Table[A368846[n, k], {n,0,10}, {k,0,n}] (* _Paolo Xausa_, Jan 08 2024 *)
%o A368846 (SageMath)
%o A368846 def A368846(n, k):
%o A368846     if k == 0: return k^n
%o A368846     if k  > n: return 0
%o A368846     return (-1)^(n + k)*2*binomial(2*k - 1, n)*binomial(2*n + 1, 2*k)
%o A368846 for n in range(10): print([A368846(n, k) for k in range(n+1)])
%Y A368846 Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.
%K A368846 sign,tabl
%O A368846 0,3
%A A368846 _Peter Luschny_, Jan 07 2024