A395490 Triangle read by rows: coefficients of reciprocal polynomials refining A367255.

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 193, 65, 1, 1, 216, 1511, 1511, 216, 1, 1, 719, 10443, 23293, 10443, 719, 1, 1, 2423, 67733, 290291, 290291, 67733, 2423, 1, 1, 8285, 425065, 3198397, 6022529, 3198397, 425065, 8285, 1

Offset

0,5

Comments

Define an integer-valued polynomial p_n(x) = Product_{i=2..n-1} ((n + 1)*x + i) * ((3*n + 1)*(x + x^2) + n) / n!. Then the n-th row describes the numerator of the generating function Sum_{k >= 0} p_n(k) t^k as a rational function.

The cases when n = 0, 1 are a bit special, involving some cancellations.

Links

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

Formula

Numerator of the generating function of values of p_n(x) = Product_{i=2..n-1} ((n + 1) * x + i) * ((3*n + 1) * (x + x^2) + n) / n!.

Example

The first few rows are:
  1;
  1, 1;
  1, 5, 1;
  1, 19, 19, 1;
  1, 65, 193, 65, 1;
  1, 216, 1511, 1511, 216, 1;

Prog

(SageMath)
x = polygen(QQ, 'x')
t = x.parent()[['t']].gen()
def row(n):
    if n < 2: return [1] * (n + 1)
    pn = prod((n+1)*x+i for i in range(2, n)) * ((3*n+1)*(x+x**2)+n) / factorial(n)
    poly = (1 - t)**(n + 1) * sum(pn(k) * t**k for k in range(n + 1)).O(n + 1)
    return poly.coefficients()

Crossrefs

Row sums are A367255.

Similar to A108267, A395571, A394704.

Sequence in context: A029847 A154334 A178346 * A168551 A262307 A383753

Adjacent sequences: A395487 A395488 A395489 * A395491 A395492 A395493

Keyword

nonn,tabl

Author

F. Chapoton, Apr 25 2026

Status

approved