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A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx. 2
1, 1, 5, 65, 1405, 42505, 1653125, 78578225, 4414067725, 286099718425, 21015972365525, 1725374840578625, 156560122048892125, 15559151967183795625, 1680744724811088153125, 196083244062052339084625, 24570430118524659881918125, 3291153805391398126661325625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

E.g.f.: 1 + Series_Reversion( 5*log(1+x) - 4*x ).

E.g.f.: -5/4*LambertW(-4/5*exp((x-4)/5)).

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.

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.

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1405*x^4/4! + 42505*x^5/5! +...

where A(5*log(1+x) - 4*x) = 1+x.

Related series:

A(x)^5 = 1 + 5*x + 45*x^2/2! + 685*x^3/3! + 15645*x^4/4! + 485645*x^5/5! +...

1/A(x)^5 = 1 - 5*x + 5*x^2/2! - 85*x^3/3! - 1595*x^4/4! - 50645*x^5/5! +...

(d/dx Series_Reversion(Integral 1/A(x)^5 dx))^(1/5) begins:

G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 +...+ A002294(n)*x^n +...

where G(x) = 1 + x*G(x)^5.

MATHEMATICA

CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

PROG

(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)}

for(n=0, 25, print1(a(n), ", "))

(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)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* O.g.f. continued fraction: */

{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)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A006351, A058562, A234294.

Sequence in context: A079482 A147625 A157097 * A251575 A277347 A276755

Adjacent sequences:  A234292 A234293 A234294 * A234296 A234297 A234298

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 25 2013

STATUS

approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)