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

1, 3, 6, 21, 60, 174, 537, 1596, 4776, 14358, 43053, 129126, 387438, 1162272, 3486678, 10460307, 31380756, 94141830, 282426288, 847278282, 2541833808, 7625503749, 22876509444, 68629525032, 205888582014, 617665741140, 1852997213508, 5558991660912, 16676974967991, 50030924873862, 150092774683998

Offset

0,2

Comments

Compare to: exp( Sum_{n>=1} x^n/(1+x^n)/n ) = Sum_{n>=0} x^(n*(n+1)/2).

Links

Table of n, a(n) for n=0..30.

Formula

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.

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)), ...

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

Example

G.f.: A(x) = 1 + 3*x + 6*x^2 + 21*x^3 + 60*x^4 + 174*x^5 + 537*x^6 +...
such that
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)) +...

Mathematica

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 *)

Prog

(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))}
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 - 2*x^(n+1-i)*A+ x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A165941, A259274, A259275, A259276.

Sequence in context: A148621 A148622 A148623 * A054878 A084567 A294527

Adjacent sequences: A259270 A259271 A259272 * A259274 A259275 A259276

Keyword

nonn

Author

Paul D. Hanna, Jun 23 2015

Status

approved