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%I #18 Nov 16 2017 15:22:33
%S 1,10,315,16520,1212775,114465780,13204213435,1800094703440,
%T 283154358503295,50478562633826300,10057594831485171355,
%U 2214859039031666012760,534202513174577053611415,140048168049127802257998820,39652657811418543065286846075,12058716801545122639605896216480
%N E.g.f.: Series_Reversion( 6*x - 5*x*exp(x) ).
%H G. C. Greubel, <a href="/A259066/b259066.txt">Table of n, a(n) for n = 1..295</a>
%F O.g.f.: x * Sum_{n>=0} 5^n / (6 - n*x)^(n+1).
%F E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^n / n!.
%F E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^(n-1) / n! ).
%F a(n) ~ (c/(6*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(6*exp(1)/5). - _Vaclav Kotesovec_, Jun 19 2015
%e E.g.f.: A(x) = x + 10*x^2/2! + 315*x^3/3! + 16520*x^4/4! + 1212775*x^5/5! +...
%e where A(6*x - 5*x*exp(x)) = x.
%e Also we have the related infinite series.
%e O.g.f.: F(x) = x + 10*x^2 + 315*x^3 + 16520*x^4 + 1212775*x^5 +...
%e where F(x)/x = 1/6 + 5/(6-x)^2 + 5^2/(6-2*x)^3 + 5^3/(6-3*x)^4 + 5^4/(6-4*x)^5 +...
%t Rest[CoefficientList[InverseSeries[Series[6*x - 5*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jun 19 2015 *)
%o (PARI) {a(n) = local(A=x); A = serreverse(6*x - 5*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
%o for(n=1,20,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A053492, A259063, A259064, A259065.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 17 2015
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