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A368846
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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.
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6
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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[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]
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MATHEMATICA
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A368846[n_, k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];
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PROG
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(SageMath)
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)])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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