A362995 Triangle read by rows. T(n, k) = [x^k] lcm({i + 1 : 0 <= i <= n}) * (Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1)).

1, 3, 2, 11, 28, 18, 25, 184, 351, 192, 137, 2608, 11097, 16128, 7500, 147, 6816, 57591, 166912, 193750, 77760, 1089, 118464, 1865511, 9588736, 20843750, 20062080, 7058940, 2283, 567936, 16015401, 136921088, 495546875, 858003840, 704129265, 220200960

Offset: 0

Author

Peter Luschny, May 14 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
[0]    1;
[1]    3,      2;
[2]   11,     28,       18;
[3]   25,    184,      351,       192;
[4]  137,   2608,    11097,     16128,      7500;
[5]  147,   6816,    57591,    166912,    193750,     77760;
[6] 1089, 118464,  1865511,   9588736,  20843750,  20062080,   7058940;
[7] 2283, 567936, 16015401, 136921088, 495546875, 858003840, 704129265, 220200960;
		

Links

• Peter Luschny, An integer triangle for representing the Bernoulli numbers, Mathematics Stack Exchange, May 2023.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A362993 (row sums), A362994 (alternating row sums), A001008 (column 0), A362992 (main diagonal), A362996/A362997.

Cf. A363000.

Programs

Maple

R := (n, x) -> add(add(x^j*binomial(u, j)*(j + 1)^n, j = 0..u)/(u + 1), u=0..n):
CoeffList := p -> PolynomialTools:-CoefficientList(p, x):
poly := (n, x) -> ilcm(seq(i, i = 1..n+1)) * R(n, x):
seq(print(CoeffList(poly(n, x))), n = 0..7);

SageMath

def A362995row(n: int) -> list[int]:
    s = add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
        for j in (0..u)) for u in (0..n))
    l = lcm(i + 1 for i in (0..n))
    return (s * l).list()
for n in (0..7): print(A362995row(n))

Formula

T(n, k) = lcm(1,2, ..., n+1) * A362996(n, k) / A362997(n, k).

Sum_{k=0..n} (-1)^k * T(n, k) = lcm(1,2, ..., n+1) * Bernoulli(n, 1) = A362994(n).