A036675 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x^2).

1, 1, 2, 6, 18, 59, 198, 690, 2450, 8878, 32632, 121518, 457262, 1736526, 6646340, 25613086, 99298674, 387021728, 1515594560, 5960406102, 23530528512, 93216984177, 370450977206, 1476458287082, 5900150928510, 23635544130948

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000621, A101913

Programs

Maple

A := 1; f := proc(n) global A; coeff(series( 1+x*(A*subs(x=x^2,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;

Mathematica

terms = 26; A[] = 0; Do[A[x] = 1 + x*A[x]^2*A[x^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)

Maxima

T(n,m):=if m=n then 1 else sum((m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
makelist(T(2*n+1,1),n,0,30); /* Vladimir Kruchinin, Mar 18 2015 */

PARI

a(n)=local(A,m); if(n<0,0,m=2; A=1+O(x); while(m<=n+1,m*=2; A=2/(1+sqrt(1-4*x*subst(A,x,x^2)))); polcoeff(A,n))

Formula

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2/(1-x^2)) (continued fraction); more generally g.f. C(x/(1-x^2/(1-x^2))) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011]

a(n) ~ c * d^n / n^(3/2), where d = 4.250770453055989899189676464071962617426..., c = 0.600960911911396921862654605015399962... . - Vaclav Kotesovec, Aug 10 2014

a(n) = T(2*n+1,1), where T(n,m) = sum(i=1..n-m, (m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i)), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015