A052825 A simple grammar: partial sums of A008965.

0, 0, 1, 3, 6, 11, 18, 31, 50, 85, 144, 251, 438, 789, 1420, 2601, 4792, 8907, 16618, 31219, 58814, 111301, 211180, 401925, 766648, 1465899, 2808082, 5389509, 10360576, 19948155, 38460946, 74253513, 143527180, 277746975, 538048150, 1043342277, 2025049108

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..2500
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 790

Programs

Maple

spec := [S,{B=Cycle(C),C=Sequence(Z,1 <= card),S=Prod(C,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
h := n -> add(numtheory:-phi(j)/j*log((x^j-1)/(2*x^j-1)), j=1..n):
seq(coeff(series((x/(1-x))*h(n),x,n+1),x,n),n=0..36); # Peter Luschny, Oct 25 2015

Mathematica

m = 40;
gf = (x/(1-x))*Sum[EulerPhi[j]/j*Log[(x^j-1)/(2*x^j-1)], {j,1,m}] + O[x]^m;
CoefficientList[gf, x] (* Jean-François Alcover, Jun 03 2019 *)

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

Formula

G.f.: (x/(x-1))*Sum_{j>=1} (A000010(j)/j)*log((x^j-1)/(2*x^j-1)).

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

Extensions

More terms from Danny Rorabaugh, Oct 25 2015