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G.f. satisfies: 6*A(x) = 5 + x + A(x)^5, starting with [1,1,10].
2

%I #7 Nov 28 2017 03:41:16

%S 1,1,10,210,5505,161601,5082420,167451780,5705082795,199354509755,

%T 7105393162010,257312347583330,9440808323869455,350189693739455535,

%U 13110655796699158800,494772468434359266960,18801468275832345890970

%N G.f. satisfies: 6*A(x) = 5 + x + A(x)^5, starting with [1,1,10].

%C See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

%F G.f.: A(x) = 1 + Series_Reversion(1+6*x - (1+x)^5). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (5+x)^(4*n+1)/6^(5*n+1). - _Paul D. Hanna_, Jan 24 2008

%F a(n) ~ (-5 + 4*(6/5)^(5/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Nov 28 2017

%e A(x) = 1 + x + 10*x^2 + 210*x^3 + 5505*x^4 + 161601*x^5 +...

%e A(x)^5 = 1 + 5*x + 60*x^2 + 1260*x^3 + 33030*x^4 + 969606*x^5 +...

%t CoefficientList[1 + InverseSeries[Series[1+6*x - (1+x)^5, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)

%o (PARI) {a(n)=local(A=1+x+10*x^2+x*O(x^n));for(i=0,n,A=A-6*A+5+x+A^5);polcoeff(A,n)}

%Y Cf. A120588 - A120595, A120597 - A120607.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2006

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