A218250 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2.
1, 1, 3, 7, 18, 49, 135, 383, 1104, 3228, 9554, 28557, 86095, 261487, 799323, 2457327, 7592620, 23565444, 73437284, 229691620, 720800824, 2268820824, 7161255962, 22661307317, 71878917199, 228487568175, 727779875401, 2322485254421, 7424488376794, 23773398866825
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 135*x^6 + 383*x^7 +... where A(x) = 1 + (1+2*x)*x*A(x) + (2+x)*x^3*A(x)^2 + x^5*A(x)^3.
Programs
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Mathematica
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) * (1 + x^2*AGF)^2 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *) -
PARI
{a(n)=local(A=1); for(i=1, n, A=(1+x*A)*(1+x^2*A)^2+x*O(x^n)); polcoeff(A, n)} for(n=0,30,print1(a(n),", "))
Formula
Recurrence: 2*(n+2)*(2*n+5)*(43*n^3 - 48*n^2 - 43*n + 12)*a(n) = 2*(2*n+1)*(2*n+3)*(43*n^3 - 5*n^2 - 94*n + 8)*a(n-1) + 2*(344*n^5 + 132*n^4 - 1303*n^3 - 399*n^2 + 554*n + 168)*a(n-2) + (473*n^5 - 528*n^4 - 1711*n^3 + 1866*n^2 + 1256*n - 960)*a(n-3) - 6*(86*n^5 - 225*n^4 - 321*n^3 + 794*n^2 + 160*n - 416)*a(n-4) + 4*(n-4)*(n-2)*(43*n^3 + 81*n^2 - 10*n - 36)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.361963061296269297... is the root of the equation -4 + 12*d - 11*d^2 - 16*d^3 - 8*d^4 + 4*d^5 = 0 and c = 2.227460242885392531198808525530878354... - Vaclav Kotesovec, Sep 10 2013