A369135 -id:A369135 - cp's OEIS frontend

A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)L(n) M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

-1, 0, 1, 0, 0, -1, 0, 0, 7, 3, 0, 0, -14, -6, -1, 0, 0, 693, 297, 55, 5, 0, 0, -30030, -12870, -2431, -260, -15, 0, 0, 4150146, 1778634, 337480, 37310, 2625, 105, 0, 0, -21441420, -9189180, -1745458, -194480, -14280, -714, -21

Offset: 0

Peter Luschny, Jan 14 2024

As has been observed by T. Curtright, the absolute value of the nonzero terms in row n of the triangle is monotonically decreasing, and the absolute value of each nonzero term T(n, k) is greater than the sum of the absolute value of the terms in the tail of that row.

The sum of the n-th row divided by the lcm of the n-th row of A368848 is the Bernoulli number B(2*n).

			[0] [-1]
[1] [0, 1]
[2] [0, 0,        -1]
[3] [0, 0,         7,        3]
[4] [0, 0,       -14,       -6,       -1]
[5] [0, 0,       693,      297,       55,       5]
[6] [0, 0,    -30030,   -12870,    -2431,    -260,    -15]
[7] [0, 0,   4150146,  1778634,   337480,   37310,   2625,  105]
[8] [0, 0, -21441420, -9189180, -1745458, -194480, -14280, -714, -21]
.
For n = 5:
(0 + 0 + 693 + 297 + 55 + 5) / 13860 = 5 / 66 = Bernoulli(10).

Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.

Cf. A368846, A368848, A369135, A369120 (row sums), A369121 (T(n,n)), A369122 (T(n,2)), A000367/A002445 (Bernoulli(2n)).

A368846[n_, k_] := If[k == 0, Boole[n == 0], (-1)^(n + k) 2 Binomial[2 k - 1, n] Binomial[2 n + 1, 2 k]];
Map[# LCM @@ Denominator[#]&, MapIndexed[(-1)^First[#2] Take[#, First[#2]]&, Inverse[PadRight[Table[A368846[n, k], {n, 0, 10}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 15 2024 *)

M = matrix(ZZ, 32, 32, A368846).inverse()
def T(n, k):
    L = (-1)**(n + 1)*lcm(M[n][k].denominator() for k in range(n + 1))
    return L * M[n][k]
for n in range(9):
    print([T(n, k) for k in range(n + 1)])

(Sum_{k=0..n} T(n, k)) / A369135(n) = Bernoulli(2*n).

T(n, 2) / T(n, 3) = 7 / 3 for n >= 3.

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.

Original entry on oeis.org

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)])

A368847 Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 1, 17, 1, 1, 0, 0, 691, 691, 59, 41, 5, 1, 0, 0, 14, 2, 359, 8, 4, 1, 1, 0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1, 0, 0, 43867, 43867, 750167, 6583, 943, 1129, 217, 2, 1, 0, 0, 1222277, 174611, 627073, 1540967, 28399, 53, 47, 23, 1, 1

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).

			[0] [1]
[1] [0, 1]
[2] [0, 0,    1]
[3] [0, 0,    1,     1]
[4] [0, 0,    1,     1,    1]
[5] [0, 0,    1,     3,    1,   1]
[6] [0, 0,    1,     1,   17,   1,   1]
[7] [0, 0,  691,   691,   59,  41,   5, 1]
[8] [0, 0,   14,     2,  359,   8,   4, 1, 1]
[9] [0, 0, 3617, 10851, 1237, 217, 293, 1, 7, 1]

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.
Peter Luschny, Illustrating the polynomials.

Cf. A368846 (inverse), A368848 (denominator), A369134, A369135, A000367/A002445 (Bernoulli(2n)).

A368846[n_,k_]:=If[k==0,Boole[n==0],(-1)^(n+k)2Binomial[2k-1,n]Binomial[2n+1,2k]];
Numerator[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].numerator() for k in range(n+1)])

A369120 a(n) = Sum_{k=0..n} A369134(n, k).

Original entry on oeis.org

-1, 1, -1, 10, -21, 1050, -45606, 6306300, -32585553, 5332033850, -646674981498, 248655336029100, -579463114572870, 400708622091878100, -31790012531476579380, 43435207772044760997000, -31514892593265599765292045, 814247185935070977732893250

Offset: 0

Peter Luschny, Jan 14 2024

sign

See A369134 for comments.

Cf. A369134, A000367/A002445 (Bernoulli(2n)).

A368846[n_, k_] := If[k == 0, Boole[n == 0], (-1)^(n + k) 2 Binomial[2 k - 1, n] Binomial[2 n + 1, 2 k]];
Map[Total[# LCM @@ Denominator[#]]&, MapIndexed[(-1)^First[#2] Take[#, First[#2]]&, Inverse[PadRight[Table[A368846[n, k], {n, 0, 20}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 16 2024 *)

a(n) / A369135(n) = Bernoulli(2*n).

cp's OEIS Frontend

A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)*L(n) * M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

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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.

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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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A368847 Triangle read by rows: T(n, k) = numerator(M(n, k)) where M is the inverse matrix of A368846.

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A369120 a(n) = Sum_{k=0..n} A369134(n, k).

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A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)L(n) M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

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.