A393141 - OEIS

%I #9 Mar 17 2026 14:33:08

%S 1,3,17,71,1569,899,355081,425331,16541017,5719087,99920609601,

%T 144619817,98139640241473,485223422289,21844512889051,

%U 2648261961071387,236389784118231290049,458182173298217,536484538620663729658993,3387894135040576041

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

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

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

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

%p P := n -> hypergeom([1, -n], [], -1/(n + 1)):

%p seq(numer(simplify(P(n))), n = 0..19);

%o (Python)

%o from math import gcd

%o def A393141(n: int) -> int:

%o     t_num, t_den = 1, 1   # term

%o     s_num, s_den = 0, 1   # sum

%o     for k in range(n + 1):

%o         if k > 0:

%o             t_num *= (n - k + 1)

%o             t_den *= (n + 1)

%o             g = gcd(t_num, t_den)

%o             t_num //= g

%o             t_den //= g

%o         num = s_num * t_den + t_num * s_den

%o         den = s_den * t_den

%o         g = gcd(num, den)

%o         s_num, s_den = num // g, den // g

%o     return s_num

%o L = [A393141(n) for n in range(20)]; print(L)

%o (SageMath)

%o def a(n: int) -> int:

%o     s = sum(falling_factorial(n, k) / (n + 1) ** k for k in range(n + 1))

%o     return s.numerator()

%o print([a(n) for n in range(20)])

%Y Cf. A036505 (denominators), A001865, A063169.

%K nonn,frac

%O 0,2

%A _Peter Luschny_, Mar 09 2026