OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..500
FORMULA
O.g.f.: A(x) = 1 + (x+x^2)*A(x)^2 + x^3*A(x)^3 [From Simon Plouffe].
a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+1,n-k)/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-2*k+j+m,n-k)*m/(2*n-2*k+j+m) * C(n-k,k-j)*C(k-j,j).
a(n) ~ s * sqrt((1 - r*s)/(Pi*(1 - r*s*(1 + r)))) / (2*n^(3/2)*r^n), where r = (1 + 5*sqrt(5))/62 = 0.1964570949596604593878365861880061480194047064202... and s = 4*sqrt(5)-7 = 1.944271909999158785636694674925104941762473438446... are real roots of the system of equations 1 + r*(1 + r)*s^2 + r^3*s^3 = s, r*s*(2 + 2*r + 3*r^2*s) = 1. - Vaclav Kotesovec, Nov 22 2017, updated Jul 19 2021
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + 139*x^5 + 560*x^6 +...
A(x) = Catalan(x + x^2 + x^3*A(x)) where:
Catalan(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
MATHEMATICA
m = 24; Catala[x_] = (1 - Sqrt[1 - 4 x])/(2 x); A[_] = 0;
Do[A[x_] = Catala[x + x^2 + x^3 A[x]] + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 22 2019 *)
PROG
(PARI) {a(n, m=1)=sum(k=0, n, sum(j=0, k, binomial(2*n-2*k+j+m, n-k)*m/(2*n-2*k+j+m)*binomial(n-k, k-j)*binomial(k-j, j)))}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=2/(1+sqrt(1-4*(x+x^2 +x^3*A)))); polcoeff(A^m, n)}
(PARI) {a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+(x+x^2)*A^2+x^3*A^3); polcoeff(A^m, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 26 2009
STATUS
approved