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%I #8 Apr 18 2024 11:21:55
%S 1,5,20,105,520,2580,12945,64680,323320,1616780,8083745,40418380,
%T 202092620,1010462480,5052310420,25261556205,126307777920,
%U 631538879180,3157694416720,15788472066780,78942360284720,394711801527505,1973559007551520,9867795037511480,49338975188073020
%N G.f.: A(x) = exp( Sum_{n>=1} 5^n * x^n/(n*(1+x^n)) ).
%C Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).
%F G.f.: -1/4 + (5/4)/(1+x - 5*x/(1+x^2 - 5*x^2/(1+x^3 - 5*x^3/(1+x^4 - 5*x^4/(1+x^5 - 5*x^5/(1+x^6 - 5*x^6/(1+x^7 - 5*x^7/(1+x^8 - 5*x^8/(...))))))))), a continued fraction.
%F G.f.: A(x) = (1 + x*B(x))/(1 - 4*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 4*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 4*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 4*x^4*E(x)), ...
%F a(n) ~ c * 5^n, where c = 2 / (5^(1/8) * EllipticTheta(2, 0, 1/sqrt(5))) = 0.8277706439469762656495798472679610454060848013727259... - _Vaclav Kotesovec_, Oct 18 2020, updated Apr 18 2024
%e G.f.: A(x) = 1 + 5*x + 20*x^2 + 105*x^3 + 520*x^4 + 2580*x^5 +...
%e such that
%e log(A(x)) = 5*x/(1+x) + 5^2*x^2/(2*(1+x^2)) + 5^3*x^3/(3*(1+x^3)) + 5^4*x^4/(4*(1+x^4)) + 5^5*x^5/(5*(1+x^5)) +...
%t nmax = 30; CoefficientList[Series[Exp[Sum[5^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 18 2020 *)
%o (PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 5^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 4*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A165941, A259273, A259274, A259276.
%K nonn,changed
%O 0,2
%A _Paul D. Hanna_, Jun 23 2015
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