A308448 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - 2*x^k - x^(2*k)))/k.

1, 1, 3, 6, 14, 28, 64, 135, 300, 653, 1458, 3223, 7240, 16228, 36678, 83025, 188910, 430730, 985752, 2260866, 5199612, 11982591, 27673826, 64027215, 148399514, 344490100, 800886300, 1864461210, 4346031950, 10142519585, 23696421808, 55420499295, 129742683174, 304014091125

Offset: 1

Author

Ilya Gutkovskiy, May 27 2019

Keywords

• nonn

Comments

Inverse Euler transform of A000129.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> combinat[fibonacci](n)+b(n, n-1):
seq(a(n), n=1..34);  # Alois P. Heinz, May 19 2022

Mathematica

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
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 *)

Formula

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

a(n) ~ (1 + sqrt(2))^n/n. - Vaclav Kotesovec, May 28 2019

"CHK" (necklace, identity, unlabeled) transform of A000045. - Alois P. Heinz, May 19 2022