A200716 - OEIS

%I #18 Sep 19 2013 11:55:07

%S 1,1,4,17,84,453,2574,15185,92119,571022,3600981,23029021,149000790,

%T 973581692,6415198045,42580369370,284427460919,1910594331920,

%U 12898153658337,87461992473577,595455441375978,4068652368270955,27891991988552554,191783482751813061,1322319472577803761

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

%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)).

%H Vaclav Kotesovec, <a href="/A200716/b200716.txt">Table of n, a(n) for n = 0..500</a>

%H Vaclav Kotesovec, <a href="/A200716/a200716.txt">Recurrence (of order 11)</a>

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

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

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

%F a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 7.342019160707096169... is the root of the equation -27 + 108*d^2 - 162*d^4 + 54*d^5 + 108*d^6 + 216*d^7 - 27*d^8 - 18*d^9 - 27*d^10 + 4*d^11 = 0 and c = 0.468554406193087607276981923311829947714908080994... - _Vaclav Kotesovec_, Sep 19 2013

%e G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 84*x^4 + 453*x^5 + 2574*x^6 +...

%e Related expansions:

%e A(x)^3 = 1 + 3*x + 15*x^2 + 76*x^3 + 414*x^4 + 2370*x^5 + 14047*x^6 +...

%e A(x)^4 = 1 + 4*x + 22*x^2 + 120*x^3 + 685*x^4 + 4048*x^5 + 24558*x^6 +...

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

%e The logarithm of the g.f. A = A(x) equals the series:

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

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

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

%e (1 + 5^2*x/A^2 + 10^2*x^2/A^4 + 10^2*x^3/A^6 + 5^2*x^4/A^8 + x^5/A^10)*x^5*A^10/5 + ...

%t nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF^3) * (1 + x^2*AGF) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 19 2013 *)

%o (PARI) {a(n)=local(p=2,q=-2,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=-2,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=-2,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. A200717, A200718, A200719, A200074, A200075, A199874, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 21 2011