A200716 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)).
1, 1, 4, 17, 84, 453, 2574, 15185, 92119, 571022, 3600981, 23029021, 149000790, 973581692, 6415198045, 42580369370, 284427460919, 1910594331920, 12898153658337, 87461992473577, 595455441375978, 4068652368270955, 27891991988552554, 191783482751813061, 1322319472577803761
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 84*x^4 + 453*x^5 + 2574*x^6 +... Related expansions: A(x)^3 = 1 + 3*x + 15*x^2 + 76*x^3 + 414*x^4 + 2370*x^5 + 14047*x^6 +... A(x)^4 = 1 + 4*x + 22*x^2 + 120*x^3 + 685*x^4 + 4048*x^5 + 24558*x^6 +... where A(x) = 1 + x*A(x)^3 + x^2*A(x) + x^3*A(x)^4. The logarithm of the g.f. A = A(x) equals the series: 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 + (1 + 3^2*x/A^2 + 3^2*x^2/A^4 + x^3/A^6)*x^3*A^6/3 + (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 + (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 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..500
- Vaclav Kotesovec, Recurrence (of order 11)
Crossrefs
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^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 *) -
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)} -
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)} -
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)}
Formula
G.f. A(x) satisfies:
(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)] ).
(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)] ).
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
Comments