A368847 -id:A368847 - cp's OEIS frontend

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) 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

Peter Luschny, Jan 07 2024

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.

			[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]

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.

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

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 *)

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)])

A368848 Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.

Original entry on oeis.org

1, 1, 6, 1, 1, 30, 1, 1, 60, 140, 1, 1, 45, 105, 630, 1, 1, 20, 140, 252, 2772, 1, 1, 6, 14, 1260, 693, 12012, 1, 1, 900, 2100, 945, 5940, 10296, 51480, 1, 1, 3, 1, 945, 189, 1287, 6435, 218790, 1, 1, 100, 700, 420, 660, 12012, 780, 145860, 923780

Offset: 0

Peter Luschny, Jan 07 2024

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).

			Triangle starts:
[0] [1]
[1] [1, 6]
[2] [1, 1,  30]
[3] [1, 1,  60,  140]
[4] [1, 1,  45,  105,  630]
[5] [1, 1,  20,  140,  252, 2772]
[6] [1, 1,   6,   14, 1260,  693, 12012]
[7] [1, 1, 900, 2100,  945, 5940,  10296, 51480]
[8] [1, 1,   3,    1,  945,  189,   1287,  6435, 218790]

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.

Cf. A368846 (inverse), A368847 (numerator), A002457 (main diagonal), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

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

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].denominator() for k in range(n+1)])

cp's OEIS Frontend

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

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A368848 Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.

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cp's OEIS Frontend

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.

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A368848 Triangle read by rows: T(n, k) = denominator(M(n, k)) where M is the inverse matrix of A368846.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

SageMath

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