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
Examples
1, -1/2, 2/3, -3/2, 24/5, -20, 720/7, -630, 4480, -36288, 3628800/11, -3326400, 479001600/13, -444787200, ...
Links
- 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.
Programs
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Mathematica
a[n_]:=Numerator[(-1)^n*n!/(n+1)]; Array[a,24,0] (* Stefano Spezia, Jun 24 2024 *)
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PARI
a(n) = numerator(sum(i=0, n, stirling(n, i, 1)*bernfrac(i)));
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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
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