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A052825 A simple grammar: partial sums of A008965.

%I A052825 #34 Jan 13 2025 08:29:59
%S A052825 0,0,1,3,6,11,18,31,50,85,144,251,438,789,1420,2601,4792,8907,16618,
%T A052825 31219,58814,111301,211180,401925,766648,1465899,2808082,5389509,
%U A052825 10360576,19948155,38460946,74253513,143527180,277746975,538048150,1043342277,2025049108
%N A052825 A simple grammar: partial sums of A008965.
%H A052825 Danny Rorabaugh, <a href="/A052825/b052825.txt">Table of n, a(n) for n = 0..2500</a>
%H A052825 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=790">Encyclopedia of Combinatorial Structures 790</a>
%F A052825 G.f.: (x/(x-1))*Sum_{j>=1} (A000010(j)/j)*log((x^j-1)/(2*x^j-1)).
%F A052825 a(n) ~ 2^n/n * (1 + 2/n + 6/n^2 + 26/n^3 + 150/n^4 + 1082/n^5 + 9366/n^6 + 94586/n^7 + 1091670/n^8 + 14174522/n^9 + 204495126/n^10 + ...), for coefficients see A000629. - _Vaclav Kotesovec_, Jun 03 2019
%p A052825 spec := [S,{B=Cycle(C),C=Sequence(Z,1 <= card),S=Prod(C,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052825 h := n -> add(numtheory:-phi(j)/j*log((x^j-1)/(2*x^j-1)), j=1..n):
%p A052825 seq(coeff(series((x/(1-x))*h(n),x,n+1),x,n),n=0..36); # _Peter Luschny_, Oct 25 2015
%t A052825 m = 40;
%t A052825 gf = (x/(1-x))*Sum[EulerPhi[j]/j*Log[(x^j-1)/(2*x^j-1)], {j,1,m}] + O[x]^m;
%t A052825 CoefficientList[gf, x] (* _Jean-François Alcover_, Jun 03 2019 *)
%o A052825 (Sage) var('x'); a = lambda n: taylor(x/(1-x) * sum([taylor(euler_phi(i)/i * log((x^i - 1)/(2*x^i - 1)), x, 0, n) for i in range(1, n+1)]), x, 0, n).coefficient(x^n) # _Danny Rorabaugh_, Oct 25 2015
%K A052825 easy,nonn
%O A052825 0,4
%A A052825 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052825 More terms from _Danny Rorabaugh_, Oct 25 2015