A198953 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^3).
1, 2, 9, 56, 400, 3095, 25240, 213633, 1859006, 16527544, 149472480, 1370794835, 12718060947, 119158146283, 1125816405458, 10714275588727, 102615375322564, 988302823695146, 9565859385140272, 93000625498797314, 907782305262566776, 8892941663606408172
Offset: 0
Examples
G.f.: A(x) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 +... Related expansions. A(x)^2 = 1 + 4*x + 22*x^2 + 148*x^3 + 1105*x^4 + 8798*x^5 + 73196*x^6 +... A(x)^3 = 1 + 6*x + 39*x^2 + 284*x^3 + 2223*x^4 + 18267*x^5 + 155445*x^6 +... A(x)^4 = 1 + 8*x + 60*x^2 + 472*x^3 + 3878*x^4 + 32948*x^5 + 287300*x^6 +... where A(x) = 1 + x*(A(x) + A(x)^3) + x^2*A(x)^4. The logarithm of the g.f. equals the series: log(A(x)) = (1 + A(x)^2)*x + (1 + 2^2*A(x)^2 + A(x)^4)*x^2/2 + (1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3/3 + (1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4/4 + (1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5/5 +... more explicitly, log(A(x)) = 2*x + 14*x^2/2 + 122*x^3/3 + 1118*x^4/4 + 10557*x^5/5 + 101642*x^6/6 + 991916*x^7/7 +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x*AGF^3) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 19 2013 *) -
Maxima
a(n):=sum((sum((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1),k,0,j))*(-1)^(n-j)*binomial(2*n-j,n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */
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PARI
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(2*j))*x^m/m))); polcoeff(A, n)} -
PARI
{a(n)=polcoeff((1/x)*serreverse( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2 +x^3*O(x^n)))),n)} -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*A)*(1 + x*(A+x*O(x^n))^3));polcoeff(A,n)}
Formula
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A073157 (Schroeder n-paths containing no FFs).
The formal inverse of g.f. A(x) is (sqrt((1-x^2)^2 + 4*x^3) - (1+x^2)) / (2*x^3).
D-finite with recurrence: 2*n*(n+1)*(2*n+1)*(1275*n^5 - 11696*n^4 + 36827*n^3 - 40618*n^2 - 5828*n + 25368)*a(n) = 6*n*(2*n - 1)*(7650*n^6 - 66351*n^5 + 183953*n^4 - 102147*n^3 - 314787*n^2 + 450754*n - 137760)*a(n-1) - 6*(n-1)*(2*n - 3)*(34425*n^6 - 281367*n^5 + 690471*n^4 - 86579*n^3 - 1831014*n^2 + 2230808*n - 685440)*a(n-2) + 6*(22950*n^8 - 279378*n^7 + 1275447*n^6 - 2461807*n^5 + 518525*n^4 + 5756973*n^3 - 9486182*n^2 + 5962912*n - 1303680)*a(n-3) - 6*(22950*n^8 - 313803*n^7 + 1633059*n^6 - 3736233*n^5 + 1886879*n^4 + 7909228*n^3 - 16107824*n^2 + 11531408*n - 2756544)*a(n-4) + 3*(n-4)*(3*n - 14)*(3*n - 7)*(1275*n^5 - 5321*n^4 + 2793*n^3 + 12437*n^2 - 16992*n + 5328)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 10.5255382776611313... is the root of the equation -27 + 108*d - 108*d^2 + 324*d^3 - 72*d^4 + 4*d^5 = 0 and c = 0.5321376859604656812266678970406658537671... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1)))*(-1)^(n-j)*binomial(2*n-j,n-j)). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1). - Seiichi Manyama, Jul 19 2023
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