A362997 Triangle read by rows. T(n, k) = denominator([x^k] R(n, n, x)), where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

1, 2, 1, 6, 3, 1, 12, 3, 4, 1, 60, 15, 20, 5, 1, 20, 5, 20, 15, 6, 1, 140, 35, 140, 105, 42, 7, 1, 280, 35, 280, 105, 168, 7, 8, 1, 2520, 315, 280, 315, 504, 7, 72, 9, 1, 2520, 315, 280, 315, 504, 35, 360, 45, 10, 1, 27720, 3465, 3080, 3465, 5544, 385, 3960, 495, 110, 11, 1

Offset: 0

Author

Peter Luschny, May 13 2023

Keywords

• nonn
• tabl
• frac

Examples

			Triangle T(n, k) starts:
[0]    1;
[1]    2,   1;
[2]    6,   3,   1;
[3]   12,   3,   4,   1;
[4]   60,  15,  20,   5,   1;
[5]   20,   5,  20,  15,   6,  1;
[6]  140,  35, 140, 105,  42,  7,   1;
[7]  280,  35, 280, 105, 168,  7,   8,  1;
[8] 2520, 315, 280, 315, 504,  7,  72,  9,  1;
[9] 2520, 315, 280, 315, 504, 35, 360, 45, 10, 1;
		

Crossrefs

Cf. A362996 (numerator), A002805 (column 0), A362995.

Programs

SageMath

def R(n, k, x):
    return add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
           for j in (0..u)) for u in (0..k))
def A362997row(n: int) -> list[int]:
    return [r.denominator() for r in R(n, n, x).list()]
for n in (0..9): print(A362997row(n))

Formula

T(n, k) = lcm(1, 2, ..., n+1) * A362996(n, k) / A362995(n, k).