A343206 - OEIS

%I #22 Jun 24 2024 12:27:36

%S 1,-1,2,-3,24,-20,720,-630,4480,-36288,3628800,-3326400,479001600,

%T -444787200,5811886080,-81729648000,20922789888000,-19760412672000,

%U 6402373705728000,-6082255020441600,115852476579840000,-2322315553259520000,1124000727777607680000,-1077167364120207360000

%N Numerators of Daehee numbers.

%H Dae San Kim and Taekyun Kim, <a href="https://arxiv.org/abs/1309.2109">Daehee Numbers and polynomials</a>, arXiv:1309.2109 [math.NT], 2013.

%H Dae San Kim and Taekyun Kim, <a href="http://dx.doi.org/10.12988/ams.2013.39535">Daehee numbers and polynomials</a>, Applied Mathematical Sciences, Vol. 7, 2013, no. 120, 5969-5976.

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

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

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

%F a(n) = numerator((-1)^n*n!/(n+1)). - _Stefano Spezia_, Jun 24 2024

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

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

%o (PARI) a(n) = numerator(sum(i=0, n, stirling(n, i, 1)*bernfrac(i)));

%o (PARI) my(x='x+O('x^30), v=Vec(serlaplace(log(1+x)/x))); apply(numerator,v)

%o (Python)

%o from sympy.functions.combinatorial.numbers import stirling, bernoulli

%o 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

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

%Y Cf. A014973 (denominators).

%K sign,frac

%O 0,3

%A _Michel Marcus_, Apr 08 2021