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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)).
5
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
0,2
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).
EXAMPLE
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;
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);
PROG
(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))
CROSSREFS
Cf. A362993 (row sums), A362994 (alternating row sums), A001008 (column 0), A362996/A362997.
Cf. A363000.
Sequence in context: A358589 A087629 A254214 * A178384 A372803 A078563
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 14 2023
STATUS
approved