A165941 - OEIS

%I #30 Apr 18 2024 11:25:37

%S 1,2,2,6,10,18,42,78,154,314,626,1246,2498,4994,9970,19974,39930,

%T 79826,159706,319374,638714,1277530,2554978,5109854,10219922,20439714,

%U 40879234,81758854,163517466,327034514,654069866,1308139246,2616277578

%N G.f.: A(x) = exp( Sum_{n>=1} 2^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 + 2/(1+x - 2*x/(1+x^2 - 2*x^2/(1+x^3 - 2*x^3/(1+x^4 - 2*x^4/(1+x^5 - 2*x^5/(1+x^6 - 2*x^6/(1+x^7 - 2*x^7/(1+x^8 - 2*x^8/(...))))))))), a continued fraction.

%F G.f.: A(x) = (1 + x*B(x))/(1 - x*B(x)), where B(x) = (1 + x^2*C(x))/(1 - x^2*C(x)), C(x) = (1 + x^3*D(x))/(1 - x^3*D(x)), D(x) = (1 + x^4*E(x))/(1 - x^4*E(x)), ... - _Paul D. Hanna_, Jun 14 2015

%F a(n) ~ c * 2^n, where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846... - _Vaclav Kotesovec_, Oct 18 2020, updated Apr 18 2024

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 18*x^5 + 42*x^6 + 78*x^7 +...

%e such that

%e log(A(x)) = 2*x/(1+x) + 2^2*x^2/(2*(1+x^2)) + 2^3*x^3/(3*(1+x^3)) + 2^4*x^4/(4*(1+x^4)) + 2^5*x^5/(5*(1+x^5)) +...

%e Also, A(x) = (1 + x*B(x))/(1 - x*B(x)), where

%e B(x) = 1 + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^6 + 8*x^7 + 6*x^8 + 20*x^9 + 18*x^10 + 36*x^11 + 54*x^12 + 76*x^13 + 150*x^14 + 172*x^15 +...

%e such that B(x) = (1 + x*C(x))/(1 - x*C(x)), where

%e C(x) = 1 + 2*x^3 + 2*x^6 + 4*x^7 + 2*x^9 + 8*x^10 + 4*x^11 + 10*x^12 + 12*x^13 + 16*x^14 + 22*x^15 + 32*x^16 + 44*x^17 + 66*x^18 +...

%e such that C(x) = (1 + x*D(x))/(1 - x*D(x)), where

%e D(x) = 1 + 2*x^4 + 2*x^8 + 4*x^9 + 2*x^12 + 8*x^13 + 4*x^14 + 8*x^15 + 2*x^16 + 12*x^17 + 16*x^18 + 20*x^19 + 18*x^20 + 24*x^21 +...

%e such that D(x) = (1 + x*E(x))/(1 - x*E(x)), where

%e E(x) = 1 + 2*x^5 + 2*x^10 + 4*x^11 + 2*x^15 + 8*x^16 + 4*x^17 + 8*x^18 + 2*x^20 + 12*x^21 + 16*x^22 + 20*x^23 + 16*x^24 + 10*x^25 +...

%e such that E(x) = (1 + x*F(x))/(1 - x*F(x)), where

%e F(x) = 1 + 2*x^6 + 2*x^12 + 4*x^13 + 2*x^18 + 8*x^19 + 4*x^20 + 8*x^21 + 2*x^24 + 12*x^25 + 16*x^26 + 20*x^27 + 16*x^28 + 8*x^29 + 18*x^30 + 16*x^31 + 36*x^32 +...

%e etc.

%e The coefficients in the above functions tend toward the terms in triangle A259192.

%t nmax = 40; CoefficientList[Series[Exp[Sum[2^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, 2^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 - x^(n+1-i)*A+ x*O(x^n)));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A259192, A259273, A259274, A259275, A259276, A348902.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 20 2009