cp's OEIS Frontend

A259273 G.f.: A(x) = exp( Sum_{n>=1} 3^n * x^n/(n*(1+x^n)) ).

%I A259273 #9 Apr 18 2024 11:24:02
%S A259273 1,3,6,21,60,174,537,1596,4776,14358,43053,129126,387438,1162272,
%T A259273 3486678,10460307,31380756,94141830,282426288,847278282,2541833808,
%U A259273 7625503749,22876509444,68629525032,205888582014,617665741140,1852997213508,5558991660912,16676974967991,50030924873862,150092774683998
%N A259273 G.f.: A(x) = exp( Sum_{n>=1} 3^n * x^n/(n*(1+x^n)) ).
%C A259273 Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).
%F A259273 G.f.: -1/2 + (3/2)/(1+x - 3*x/(1+x^2 - 3*x^2/(1+x^3 - 3*x^3/(1+x^4 - 3*x^4/(1+x^5 - 3*x^5/(1+x^6 - 3*x^6/(1+x^7 - 3*x^7/(1+x^8 - 3*x^8/(...))))))))), a continued fraction.
%F A259273 G.f.: A(x) = (1 + x*B(x))/(1 - 2*x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - 2*x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - 2*x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - 2*x^4*E(x)), ...
%F A259273 a(n) ~ c * 3^n, where c = 2 / (3^(1/8) * EllipticTheta(2, 0, 1/sqrt(3))) = 0.7289909630241618243925302344904284400138198884186993... - _Vaclav Kotesovec_, Oct 18 2020, updated Apr 18 2024
%e A259273 G.f.: A(x) = 1 + 3*x + 6*x^2 + 21*x^3 + 60*x^4 + 174*x^5 + 537*x^6 +...
%e A259273 such that
%e A259273 log(A(x)) = 3*x/(1+x) + 3^2*x^2/(2*(1+x^2)) + 3^3*x^3/(3*(1+x^3)) + 3^4*x^4/(4*(1+x^4)) + 3^5*x^5/(5*(1+x^5)) +...
%t A259273 nmax = 40; CoefficientList[Series[Exp[Sum[3^k * x^k / (1 + x^k)/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 18 2020 *)
%o A259273 (PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, 3^m*x^m/(1+x^m+x*O(x^n))/m)), n))}
%o A259273 for(n=0, 30, print1(a(n), ", "))
%o A259273 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x^(n+1-i)*A)/(1 - 2*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
%o A259273 for(n=0, 30, print1(a(n), ", "))
%Y A259273 Cf. A165941, A259274, A259275, A259276.
%K A259273 nonn
%O A259273 0,2
%A A259273 _Paul D. Hanna_, Jun 23 2015