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A234295
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E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.
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2
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1, 1, 5, 65, 1405, 42505, 1653125, 78578225, 4414067725, 286099718425, 21015972365525, 1725374840578625, 156560122048892125, 15559151967183795625, 1680744724811088153125, 196083244062052339084625, 24570430118524659881918125, 3291153805391398126661325625
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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