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 A120598 G.f. satisfies: 30*A(x) = 29 + 125*x + A(x)^5, starting with [1,5,10]. 2

%I #7 Nov 28 2017 03:45:55

%S 1,5,10,90,825,8445,92820,1066740,12670635,154308775,1916370170,

%T 24177471370,309007779015,3992428316835,52059968802000,

%U 684240882022800,9055282215370050,120563388411386850,1613785688724362400

%N G.f. satisfies: 30*A(x) = 29 + 125*x + A(x)^5, starting with [1,5,10].

%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+30*x - (1+x)^5)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (29+125*x)^(4*n+1)/30^(5*n+1). - _Paul D. Hanna_, Jan 24 2008

%F a(n) ~ 5^(-1/2 + 3*n) * (-29 + 24*6^(1/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Nov 28 2017

%e A(x) = 1 + 5*x + 10*x^2 + 90*x^3 + 825*x^4 + 8445*x^5 +...

%e A(x)^5 = 1 + 25*x + 300*x^2 + 2700*x^3 + 24750*x^4 + 253350*x^5 +...

%t CoefficientList[1 + InverseSeries[Series[(1+30*x - (1+x)^5)/125, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)

%o (PARI) {a(n)=local(A=1+5*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-30*A+29+125*x+A^5)/25);polcoeff(A,n)}

%Y Cf. A120588 - A120597, A120599 - A120607.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 16 2006

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Last modified August 14 19:51 EDT 2024. Contains 375167 sequences. (Running on oeis4.)