A180305 G.f.: 1/(1 + x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

1, 1, 4, 11, 34, 96, 288, 833, 2456, 7175, 21054, 61633, 180674, 529220, 1550800, 4543446, 13312552, 39004278, 114281748, 334837511, 981059294, 2874447292, 8421986238, 24675963950, 72299290794, 211833080161, 620659794584, 1818500391218, 5328110328116, 15611082044176, 45739647180588, 134014753120706, 392656158141832

Offset: 0

Author

Paul D. Hanna, Jan 18 2011

Keywords

• nonn

Comments

INVERT transform of sigma (A000203). - Alois P. Heinz, Feb 11 2021

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 11*x^3 + 34*x^4 + 96*x^5 + 288*x^6 +...
eta(x)^3/A(x) = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 + 104*x^21 +...+ A184363(n)*x^n +...
1 + x*d/dx log(eta(x)) = 1 - x - 3*x^2 - 4*x^3 - 7*x^4 - 6*x^5 - 12*x^6 - 8*x^7 - 15*x^8 +...+ -sigma(n)*x^n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A184363, A000203, A000041.

Row sums of A319083.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[sigma](i), i=1..n))
    end:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 11 2021

Mathematica

nmax = 50; CoefficientList[Series[1/(1 - Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2017 *)

PARI

{a(n)=polcoeff(1/(1+x*deriv(log(eta(x+x*O(x^n))))), n)}

PARI

{a(n)=if(n==0,1,sum(k=0,n-1,sigma(n-k)*a(k)))}

PARI

N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1,N, x^k/(1-x^k)^2))) \\ Joerg Arndt, Mar 09 2014

Formula

a(n) = Sum_{k=0,n-1} sigma(n-k)*a(k) for n>0 with a(0) = 1.

G.f.: 1/(1 - sum(k>=1, x^k/(1-x^k)^2)). [Joerg Arndt, Mar 09 2014]

a(n) ~ c * d^n, where d = 2.92994725111235280869138453465150817383965264075630759525007993985560038385... is the root of the equation Sum_{k>=1} sigma(k)/d^k = 1 and c = 0.45133473613134383104139698267531812019856702278773719486399141396046228911... - Vaclav Kotesovec, Jul 28 2018