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%I #20 Feb 11 2021 04:55:21
%S 1,1,4,11,34,96,288,833,2456,7175,21054,61633,180674,529220,1550800,
%T 4543446,13312552,39004278,114281748,334837511,981059294,2874447292,
%U 8421986238,24675963950,72299290794,211833080161,620659794584,1818500391218,5328110328116,15611082044176,45739647180588,134014753120706,392656158141832
%N 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.
%C INVERT transform of sigma (A000203). - _Alois P. Heinz_, Feb 11 2021
%H Vaclav Kotesovec, <a href="/A180305/b180305.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0,n-1} sigma(n-k)*a(k) for n>0 with a(0) = 1.
%F G.f.: 1/(1 - sum(k>=1, x^k/(1-x^k)^2)). [_Joerg Arndt_, Mar 09 2014]
%F 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
%e G.f.: A(x) = 1 + x + 4*x^2 + 11*x^3 + 34*x^4 + 96*x^5 + 288*x^6 +...
%e 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 +...
%e 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 +...
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(a(n-i)*numtheory[sigma](i), i=1..n))
%p end:
%p seq(a(n), n=0..36); # _Alois P. Heinz_, Feb 11 2021
%t nmax = 50; CoefficientList[Series[1/(1 - Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 06 2017 *)
%o (PARI) {a(n)=polcoeff(1/(1+x*deriv(log(eta(x+x*O(x^n))))), n)}
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n-1,sigma(n-k)*a(k)))}
%o (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
%Y Cf. A184363, A000203, A000041.
%Y Row sums of A319083.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 18 2011
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