A201970 - OEIS

%I #9 Nov 18 2017 09:45:31

%S 1,1,3,4,11,53,146,366,1563,5837,16720,59917,246090,828157,2773984,

%T 11056545,41539923,142602611,536781500,2103092222,7632057148,

%U 27948758125,108960963441,413531196001,1523673835870,5833675306068,22596747340839,85028853983908,322415760345476

%N G.f. satisfies: A(x) = 1 + x*A(x)^2 + x^2/A(x)^3.

%C This sequence consists entirely of positive integers, but negative coefficients appear in G(x) where G(x) = 1 + x*G(x)^2 + x^2/G(x)^4.

%H Vaclav Kotesovec, <a href="/A201970/b201970.txt">Table of n, a(n) for n = 0..400</a>

%H Vaclav Kotesovec, <a href="/A201970/a201970.txt">Recurrence (of order 8)</a>

%F G.f. satisfies: x^2 = A(x)^4 - A(x)^3 - x*A(x)^5.

%F a(n) ~ s*sqrt((2*r + s^5) / (Pi*(6*r + s^5))) / (2*n^(3/2)*r^n), where r = 0.2481615785281025222129020972009425192154180157568... and s = 2.036459583599244077761952153978454195241402482624... are roots of the system of equations 1 + r^2/s^3 + r*s^2 = s, 2*r*s = 1 + 3*r^2/s^4. - _Vaclav Kotesovec_, Nov 18 2017

%e G.f.: A(x) = 1 + x + 3*x^2 + 4*x^3 + 11*x^4 + 53*x^5 + 146*x^6 +...

%e Related expansions begin:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 14*x^3 + 39*x^4 + 152*x^5 + 480*x^6 +...

%e 1/A(x)^3 = 1 - 3*x - 3*x^2 + 14*x^3 - 6*x^4 - 114*x^5 + 133*x^6 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^2+x^2/(A+x*O(x^n))^3);polcoeff(A,n)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 07 2011