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%I #8 Mar 17 2024 11:55:05
%S 1,4,12,52,204,804,3244,12948,51756,207108,828364,3313332,13253580,
%T 53014116,212055852,848224660,3392897772,13571588484,54286358988,
%U 217145432052,868581718860,3474326895460,13897307565804,55589230225428,222356920980972,889427683862724,3557710735299660
%N G.f.: A(x) = exp( Sum_{n>=1} 4^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.: -1/3 + (4/3)/(1+x - 4*x/(1+x^2 - 4*x^2/(1+x^3 - 4*x^3/(1+x^4 - 4*x^4/(1+x^5 - 4*x^5/(1+x^6 - 4*x^6/(1+x^7 - 4*x^7/(1+x^8 - 4*x^8/(...))))))))), a continued fraction.
%F G.f.: A(x) = (1 + x*B(x))/(1 - 3*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 3*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 3*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 3*x^4*E(x)), ...
%F a(n) ~ c * 4^n, where c = 2^(3/4) / EllipticTheta[2, 0, 1/2] = 0.789970474669932371974378022396788915338046391238667... - _Vaclav Kotesovec_, Oct 18 2020, updated Mar 17 2024
%e G.f.: A(x) = 1 + 4*x + 12*x^2 + 52*x^3 + 204*x^4 + 804*x^5 + 3244*x^6 +...
%e such that
%e log(A(x)) = 4*x/(1+x) + 4^2*x^2/(2*(1+x^2)) + 4^3*x^3/(3*(1+x^3)) + 4^4*x^4/(4*(1+x^4)) + 4^5*x^5/(5*(1+x^5)) +...
%t nmax = 30; CoefficientList[Series[Exp[Sum[4^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, 4^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 - 3*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A165941, A259273, A259275, A259276.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 23 2015
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