A362996 Triangle read by rows. T(n, k) = numerator([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, 3, 1, 11, 14, 3, 25, 46, 117, 16, 137, 652, 3699, 1344, 125, 49, 568, 19197, 41728, 19375, 1296, 363, 9872, 621837, 2397184, 2084375, 334368, 16807, 761, 23664, 5338467, 17115136, 99109375, 7150032, 6705993, 262144

Offset: 0

Author

Peter Luschny, May 13 2023

Keywords

• nonn
• tabl
• frac

Examples

			The triangle T(n, k) begins:
[0]   1;
[1]   3,     1;
[2]  11,    14,       3;
[3]  25,    46,     117,       16;
[4] 137,   652,    3699,     1344,      125;
[5]  49,   568,   19197,    41728,    19375,    1296;
[6] 363,  9872,  621837,  2397184,  2084375,  334368,   16807;
[7] 761, 23664, 5338467, 17115136, 99109375, 7150032, 6705993, 262144;
.
The first few polynomials are:
[0]      1
[1]      x   +      3/2
[2]    3*x^2 +    (14/3)*x   +      11/6
[3]   16*x^3 +   (117/4)*x^2 +     (46/3)*x   +     25/12
[4]  125*x^4 +  (1344/5)*x^3 +  (3699/20)*x^2 +   (652/15)*x   + 137/60
[5] 1296*x^5 + (19375/6)*x^4 + (41728/15)*x^3 + (19197/20)*x^2 + (568/5)*x + 49/20
		

Crossrefs

Cf. A362997 (denominator), A001008 (column 0), A000272 (main diagonal), 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 A362996row(n: int) -> list[int]:
    return [r.numerator() for r in R(n, n, x).list()]
for n in (0..7): print(A362996row(n))

Formula

T(n, k) = A362995(n, k) * A362997(n, k) / lcm(1, 2, ..., n+1).