A120594 - OEIS

%I #7 Nov 28 2017 03:38:18

%S 1,2,6,44,394,3948,42364,476120,5532714,65935804,801461012,9897836520,

%T 123840983812,1566487308344,19999112293944,257365488659376,

%U 3334967582746218,43477505482249692,569854228738577572

%N G.f. satisfies: 8*A(x) = 7 + 8*x + A(x)^4, starting with [1,2,6].

%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+8*x - (1+x)^4)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7+8*x)^(3*n+1)/8^(4*n+1). - _Paul D. Hanna_, Jan 24 2008

%F a(n) ~ 2^(-11/6 + 3*n) * (-7 + 6*2^(1/3))^(1/2 - n) / (n^(3/2) * sqrt(3*Pi)). - _Vaclav Kotesovec_, Nov 28 2017

%e A(x) = 1 + 2*x + 6*x^2 + 44*x^3 + 394*x^4 + 3948*x^5 + 42364*x^6 +...

%e A(x)^4 = 1 + 8*x + 48*x^2 + 352*x^3 + 3152*x^4 + 31584*x^5 + 338912*x^6+..

%t CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/8, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)

%o (PARI) {a(n)=local(A=1+2*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-8*A+7+8*x+A^4)/4);polcoeff(A,n)}

%Y Cf. A120588 - A120593, A120595 - A120607.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 16 2006