cp's OEIS Frontend

A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

%I A234295 #15 Dec 26 2013 13:27:38
%S A234295 1,1,5,65,1405,42505,1653125,78578225,4414067725,286099718425,
%T A234295 21015972365525,1725374840578625,156560122048892125,
%U A234295 15559151967183795625,1680744724811088153125,196083244062052339084625,24570430118524659881918125,3291153805391398126661325625
%N A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.
%F A234295 E.g.f.: 1 + Series_Reversion( 5*log(1+x) - 4*x ).
%F A234295 E.g.f.: -5/4*LambertW(-4/5*exp((x-4)/5)).
%F A234295 E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^5 dx ) )^(1/5), where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
%F A234295 O.g.f.: 1 + x/(1-x - 4*x/(1-2*x - 4*2*x/(1-3*x - 4*3*x/(1-4*x - 4*4*x/(1-...))))), a continued fraction.
%F A234295 a(n) ~ sqrt(5) * n^(n-1) / (4*exp(n)*(5*log(5)-10*log(2)-1)^(n-1/2)). - _Vaclav Kotesovec_, Dec 26 2013
%e A234295 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1405*x^4/4! + 42505*x^5/5! +...
%e A234295 where A(5*log(1+x) - 4*x) = 1+x.
%e A234295 Related series:
%e A234295 A(x)^5 = 1 + 5*x + 45*x^2/2! + 685*x^3/3! + 15645*x^4/4! + 485645*x^5/5! +...
%e A234295 1/A(x)^5 = 1 - 5*x + 5*x^2/2! - 85*x^3/3! - 1595*x^4/4! - 50645*x^5/5! +...
%e A234295 (d/dx Series_Reversion(Integral 1/A(x)^5 dx))^(1/5) begins:
%e A234295 G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 +...+ A002294(n)*x^n +...
%e A234295 where G(x) = 1 + x*G(x)^5.
%t A234295 CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x],x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 26 2013 *)
%o A234295 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^5*intformal(1/(A^5+x*O(x^n)))); n!*polcoeff(A, n)}
%o A234295 for(n=0, 25, print1(a(n), ", "))
%o A234295 (PARI) {a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(5*log(1+X) - 4*X); n!*polcoeff(A, n)}
%o A234295 for(n=0, 25, print1(a(n), ", "))
%o A234295 (PARI) /* O.g.f. continued fraction: */
%o A234295 {a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-4*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
%o A234295 for(n=0, 25, print1(a(n), ", "))
%Y A234295 Cf. A006351, A058562, A234294.
%K A234295 nonn
%O A234295 0,3
%A A234295 _Paul D. Hanna_, Dec 25 2013