A120597 - OEIS

%I #7 Nov 28 2017 03:43:30

%S 1,2,10,120,1770,29208,516180,9554640,182867970,3589443160,

%T 71861735660,1461730482160,30123451315620,627598216410480,

%U 13197173403868200,279728425129963680,5970277970921643570,128199003794219752920

%N G.f. satisfies: 9*A(x) = 8 + 8*x + A(x)^5, starting with [1,2,10].

%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+9*x - (1+x)^5)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (8+8*x)^(4*n+1)/9^(5*n+1). - _Paul D. Hanna_, Jan 24 2008

%F a(n) ~ (-1 + 9*sqrt(3)/(10*5^(1/4)))^(1/2 - n) / (3^(3/4) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Nov 28 2017

%e A(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 1770*x^4 + 29208*x^5 +...

%e A(x)^5 = 1 + 10*x + 90*x^2 + 1080*x^3 + 15930*x^4 + 262872*x^5 +...

%t CoefficientList[1 + InverseSeries[Series[(1+9*x - (1+x)^5)/8, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)

%o (PARI) {a(n)=local(A=1+2*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-9*A+8+8*x+A^5)/4);polcoeff(A,n)}

%Y Cf. A120588 - A120596, A120598 - A120607.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 16 2006