A211248 - OEIS

%I #12 Nov 22 2017 07:55:05

%S 1,1,4,20,114,703,4565,30752,212921,1505916,10833164,79018804,

%T 583062388,4344431508,32641910199,247033970128,1881402836376,

%U 14408753414558,110897147057354,857307054338476,6653979156676983,51831065993122915,405060413133136902,3175019470333290488

%N G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^4).

%C More generally, for fixed parameters p and q, if F(x) satisfies:

%C F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),

%C then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2 and q=1.

%F G.f. A(x) satisfies:

%F (1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x-x^3)^2/(1+x^2)^2 ) ).

%F (2) A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).

%F (3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(2*n+2) / (n+1).

%F (4) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k] * x^n*A(x)^(2*n)/n ).

%F (5) A(x) = exp( Sum_{n>=1} [(1-x*A(x))^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k )] * x^n*A(x)^(2*n)/n ).

%F (6) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = (1 + x*G(x)^2)*(1 + x^2*G(x)^2) is the g.f. of A199874.

%F a(n) ~ s * sqrt((1 + 2*r*s + 3*r^2*s^4) / (3*Pi*(1 + 2*r*s + 7*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.1194948955213353102456218138370139612914667337222... and s = 1.428770161302757679335810379290625953730830139744... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^4) = s, r*s^2*(3 + 4*r*s + 7*r^2*s^4) = 1. - _Vaclav Kotesovec_, Nov 22 2017

%e G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 114*x^4 + 703*x^5 + 4565*x^6 +...

%e where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).

%e Related expansions:

%e A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 522*x^4 + 3381*x^5 + 22735*x^6 +...

%e A(x)^4 = 1 + 4*x + 22*x^2 + 132*x^3 + 841*x^4 + 5588*x^5 + 38288*x^6 +...

%e A(x)^7 = 1 + 7*x + 49*x^2 + 343*x^3 + 2429*x^4 + 17430*x^5 +...

%e where A(x) = 1 + x*A(x)^3 + x^2*A(x)^4 + x^3*A(x)^7.

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + x*A(x))*x*A(x)^2 + (1 + 2^2*x*A(x) + x^2*A(x)^2)*x^2*A(x)^4/2 +

%e (1 + 3^2*x*A(x) + 3^2*x^2*A(x)^2 + x^3*A(x)^3)*x^3*A(x)^6/3 +

%e (1 + 4^2*x*A(x) + 6^2*x^2*A(x)^2 + 4^2*x^3*A(x)^3 + x^4*A(x)^4)*x^4*A(x)^8/4 +

%e (1 + 5^2*x*A(x) + 10^2*x^2*A(x)^2 + 10^2*x^3*A(x)^3 + 5^2*x^4*A(x)^4 + x^5*A(x)^5)*x^5*A(x)^10/5 +

%e (1 + 6^2*x*A(x) + 15^2*x^2*A(x)^2 + 20^2*x^3*A(x)^3 + 15^2*x^4*A(x)^4 + 6^2*x^5*A(x)^5 + x^6*A(x)^6)*x^6*A(x)^12/6 +...

%e more explicitly,

%e log(A(x)) = x + 7*x^2/2 + 49*x^3/3 + 359*x^4/4 + 2706*x^5/5 + 20767*x^6/6 +...

%t CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1 - x - x^3)^2/(1 + x^2)^2, {x, 0, 20}], x]], x] (* _Vaclav Kotesovec_, Nov 22 2017 *)

%o (PARI) {a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-x-x^3)^2/(1+x^2+x*O(x^n))^2 ) ), n)}

%o for(n=0,30,print1(a(n),", "))

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

%o (PARI) {a(n)=local(p=2, q=1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

%o (PARI) {a(n)=local(p=2, q=1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

%Y Cf. A200716, A200717, A199876, A211249, A200731, A199874, A199877.

%Y Cf. A198951, A198953, A198957, A192415, A198888, A036765.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2012