A308448 - OEIS

%I #13 May 20 2022 04:56:48

%S 1,1,3,6,14,28,64,135,300,653,1458,3223,7240,16228,36678,83025,188910,

%T 430730,985752,2260866,5199612,11982591,27673826,64027215,148399514,

%U 344490100,800886300,1864461210,4346031950,10142519585,23696421808,55420499295,129742683174,304014091125

%N Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - 2*x^k - x^(2*k)))/k.

%C Inverse Euler transform of A000129.

%H Alois P. Heinz, <a href="/A308448/b308448.txt">Table of n, a(n) for n = 1..1000</a>

%F -1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000129.

%F a(n) ~ (1 + sqrt(2))^n/n. - _Vaclav Kotesovec_, May 28 2019

%F "CHK" (necklace, identity, unlabeled) transform of A000045. - _Alois P. Heinz_, May 19 2022

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> combinat[fibonacci](n)+b(n, n-1):

%p seq(a(n), n=1..34);  # _Alois P. Heinz_, May 19 2022

%t nmax = 34; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - 2 x^k - x^(2 k))]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t nmax = 40; s = ConstantArray[0, nmax]; Do[s[[j]] = j*Fibonacci[j, 2] - Sum[s[[d]]*Fibonacci[j - d, 2], {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* _Vaclav Kotesovec_, Aug 10 2019 *)

%Y Cf. A000045, A000129, A006206, A008683, A060280, A261329.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, May 27 2019