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A032202 Sequence (a(n): n >= 1) that shifts left 2 places under the "CIK" (necklace, indistinct, unlabeled) transform and satisfies a(1) = a(2) = 1. 1

%I #31 Sep 17 2019 04:49:43

%S 1,1,1,2,3,6,10,22,41,92,193,435,963,2215,5051,11754,27375,64381,

%T 151898,360661,859149,2055804,4934428,11883930,28699336,69497354,

%U 168691424,410399073,1000486306,2443761830,5979742904,14656709518

%N Sequence (a(n): n >= 1) that shifts left 2 places under the "CIK" (necklace, indistinct, unlabeled) transform and satisfies a(1) = a(2) = 1.

%C From _Petros Hadjicostas_, Dec 30 2018: (Start)

%C a(n+2) = (1/n)*Sum_{d|n} phi(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.

%C G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (phi(n)/n)*log(1-A(x^n)).

%C The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 46*x^5 + 117*x^6 + 281*x^7 + 707*x^8 + 1717*x^9 + 4288*x^10 + 10583*x^11 + 26401*x^12 + ...

%C (End)

%C The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CIK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) + b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 5), and so on. - _Petros Hadjicostas_, Jan 01 2019

%H Andrew Howroyd, <a href="/A032202/b032202.txt">Table of n, a(n) for n = 1..200</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>

%t m = 33; a[1] = a[2] = 1; A[_] = 0;

%t Do[A[x_] = x(a[1] + x a[2] - x Sum[EulerPhi[n] Log[1-A[x^n]]/n, {n, 1, m}]) + O[x]^m // Normal, {m}];

%t CoefficientList[A[x], x] // Rest (* _Jean-François Alcover_, Sep 17 2019 *)

%o (PARI)

%o CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}

%o seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CIK(x*p, 2*i)); Vec(p+O(x^n))} \\ _Andrew Howroyd_, Jun 20 2018

%Y Cf. A032200, A032201, A108531.

%K nonn,eigen

%O 1,4

%A _Christian G. Bower_

%E Name modified by _Petros Hadjicostas_, Jan 01 2019

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)