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A075178 Denominators of expansion of 1/x+1/log(1-x).

%I A075178 #25 Jul 02 2025 19:28:53
%S A075178 2,12,12,120,20,504,168,720,180,1320,88,65520,10920,5040,720,24480,68,
%T A075178 28728,3192,39600,27720,182160,1840,1965600,163800,39312,3024,97440,
%U A075178 2320,3437280,229152,3769920,235620,42840,280,138181680,219336,35568,1872,3247200
%N A075178 Denominators of expansion of 1/x+1/log(1-x).
%C A075178 The numerator sequence is |A006232(n+1)|, n>=0.
%C A075178 |A006232(n+1)|= numerator(r(n)), n>=1, with r(n) := sum(|stirling1(n,k)|*B(k+1)/(k+1),k=1..n), n>=1 and B(n): =A027641(n)/A027642(n) (Bernoulli numbers) and stirling1(n,m)=A008275(n,m), n>=m>=1; r(0) := 1/2.
%F A075178 Denominators from e.g.f. 1/x + 1/log(1-x) (and of signed sequence from e.g.f. 1/x - 1/log(1+x)).
%F A075178 a(n) = denominator(r(n)), n>=0, with rational r(n) defined in one of the comments.
%e A075178 r(n) sequence, n>=0: 1/2, 1/12, 1/12, 19/120, 9/20, 863/504, 1375/168, 33953/720, 57281/180,...
%t A075178 With[{nn=40},Denominator[CoefficientList[Series[1/x+1/Log[1-x], {x,0,nn}] ,x] Range[0,nn]!]] (* _Harvey P. Dale_, Feb 18 2012 *)
%o A075178 (Sage)
%o A075178 def A075178_list(len):
%o A075178     f, R, C = 1, [0], [1]+[0]*len
%o A075178     for n in (1..len):
%o A075178         for k in range(n, 0, -1):
%o A075178             C[k] = -C[k-1] * k / (k + 1)
%o A075178         C[0] = -sum(C[k] for k in (1..n))
%o A075178         R.append((C[0]*f).denominator())
%o A075178         f *= n
%o A075178     return R[1:]
%o A075178 print(A075178_list(40)) # _Peter Luschny_, Feb 21 2016
%Y A075178 Cf. A006232, A075179, A006232 with A006233, A008275.
%K A075178 nonn,frac,easy
%O A075178 0,1
%A A075178 _Wolfdieter Lang_, Sep 06 2002