A259274 G.f.: A(x) = exp( Sum_{n>=1} 4^n * x^n/(n*(1+x^n)) ).
1, 4, 12, 52, 204, 804, 3244, 12948, 51756, 207108, 828364, 3313332, 13253580, 53014116, 212055852, 848224660, 3392897772, 13571588484, 54286358988, 217145432052, 868581718860, 3474326895460, 13897307565804, 55589230225428, 222356920980972, 889427683862724, 3557710735299660
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 4*x + 12*x^2 + 52*x^3 + 204*x^4 + 804*x^5 + 3244*x^6 +... such that 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)) +...
Programs
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Mathematica
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 *) -
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))} for(n=0, 30, print1(a(n), ", ")) -
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)} for(n=0, 30, print1(a(n), ", "))
Formula
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.
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)), ...
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
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