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

0,9

Comments

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).

Links

Table of n, a(n) for n=0..44.

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

Formula

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

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

Example

[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).

Mathematica

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 *)

Prog

(SageMath)
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)])

Crossrefs

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

Sequence in context: A103237 A338305 A239120 * A021582 A098459 A171535

Adjacent sequences: A369131 A369132 A369133 * A369135 A369136 A369137

Keyword

sign,tabl

Author

Peter Luschny, Jan 14 2024

Status

approved