A393141 a(n) = numerator(hypergeom([1, -n], [], -1/(n + 1))).

1, 3, 17, 71, 1569, 899, 355081, 425331, 16541017, 5719087, 99920609601, 144619817, 98139640241473, 485223422289, 21844512889051, 2648261961071387, 236389784118231290049, 458182173298217, 536484538620663729658993, 3387894135040576041

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = numerator(Gamma(n+1, n+1) * exp(n + 1) / (n + 1)^n).

a(n) = numerator(Sum_{k=0..n} n! / ((n - k)! * (n + 1)^k)).

a(n) = numerator(Sum_{k=0..n} FallingFactorial(n, k) / (n + 1)^k ).

Maple

P := n -> hypergeom([1, -n], [], -1/(n + 1)):
seq(numer(simplify(P(n))), n = 0..19);

Prog

(Python)
from math import gcd
def A393141(n: int) -> int:
    t_num, t_den = 1, 1   # term
    s_num, s_den = 0, 1   # sum
    for k in range(n + 1):
        if k > 0:
            t_num *= (n - k + 1)
            t_den *= (n + 1)
            g = gcd(t_num, t_den)
            t_num //= g
            t_den //= g
        num = s_num * t_den + t_num * s_den
        den = s_den * t_den
        g = gcd(num, den)
        s_num, s_den = num // g, den // g
    return s_num
L = [A393141(n) for n in range(20)]; print(L)
(SageMath)
def a(n: int) -> int:
    s = sum(falling_factorial(n, k) / (n + 1) ** k for k in range(n + 1))
    return s.numerator()
print([a(n) for n in range(20)])

Crossrefs

Cf. A036505 (denominators), A001865, A063169.

Sequence in context: A241776 A270231 A265919 * A317452 A128948 A342364

Adjacent sequences: A393138 A393139 A393140 * A393142 A393143 A393144

Keyword

nonn,frac

Author

Peter Luschny, Mar 09 2026

Status

approved