A343206 Numerators of Daehee numbers.

1, -1, 2, -3, 24, -20, 720, -630, 4480, -36288, 3628800, -3326400, 479001600, -444787200, 5811886080, -81729648000, 20922789888000, -19760412672000, 6402373705728000, -6082255020441600, 115852476579840000, -2322315553259520000, 1124000727777607680000, -1077167364120207360000

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Dae San Kim and Taekyun Kim, Daehee Numbers and polynomials, arXiv:1309.2109 [math.NT], 2013.

Dae San Kim and Taekyun Kim, Daehee numbers and polynomials, Applied Mathematical Sciences, Vol. 7, 2013, no. 120, 5969-5976.

Formula

D(n) = Sum_{i=0..n} Stirling1(n, i)*Bernoulli(i).

E.g.f. for D(n): log(1+x)/x.

D(n) = a(n)/A014973(n+1).

a(n) = numerator((-1)^n*n!/(n+1)). - Stefano Spezia, Jun 24 2024

Example

1, -1/2, 2/3, -3/2, 24/5, -20, 720/7, -630, 4480, -36288, 3628800/11, -3326400, 479001600/13, -444787200, ...

Mathematica

a[n_]:=Numerator[(-1)^n*n!/(n+1)]; Array[a, 24, 0] (* Stefano Spezia, Jun 24 2024 *)

Prog

(PARI) a(n) = numerator(sum(i=0, n, stirling(n, i, 1)*bernfrac(i)));
(PARI) my(x='x+O('x^30), v=Vec(serlaplace(log(1+x)/x))); apply(numerator, v)
(Python)
from sympy.functions.combinatorial.numbers import stirling, bernoulli
def A343206(n): return sum(stirling(n, i, signed=True)*bernoulli(i) for i in range(n+1)).p # Chai Wah Wu, Apr 08 2021

Crossrefs

Cf. A008275 (Stirling1), A027641/A027642 (Bernoulli).

Cf. A014973 (denominators).

Sequence in context: A160606 A099617 A092043 * A055067 A037319 A032811

Adjacent sequences: A343203 A343204 A343205 * A343207 A343208 A343209

Keyword

sign,frac

Author

Michel Marcus, Apr 08 2021

Status

approved