OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f.: A(x) = 1 + x*A(x)*(1+A(x)+A(x)^2) + x^2*A(x)^3*(1+A(x)+A(x)^2) + x^3*A(x)^6.
a(n) ~ sqrt((s*(1 + s + (1 + 2*r)*s^2 + 2*r*s^3 + 2*r*s^4 + 3*r^2*s^5)) / (1 + 3*(1 + r)*s + 6*r*s^2 + 10*r*s^3 + 15*r^2*s^4)) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.06228198686712455165459532624572875420874352588006064829276... and s = 1.61944833450852965640457413211207525783408084239130679443147... are roots of the system of equations (1 + r*s) * (1 + r*s^2) * (1 + r*s^3) = s, r*(1 + 2*s + 3*(1+r)*s^2 + 4*r*s^3 + 5*r*s^4 + 6*r^2*s^5) = 1. - Vaclav Kotesovec, Aug 24 2017
EXAMPLE
G.f.: A(x) = 1 + 3*x + 21*x^2 + 199*x^3 + 2166*x^4 + 25551*x^5 +..
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 524*x^3 + 5967*x^4 + 72456*x^5 +...
A(x)^3 = 1 + 9*x + 90*x^2 + 1002*x^3 + 11970*x^4 + 150057*x^5 +...
A(x)^4 = 1 + 12*x + 138*x^2 + 1660*x^3 + 20823*x^4 + 269964*x^5 +...
A(x)^5 = 1 + 15*x + 195*x^2 + 2525*x^3 + 33255*x^4 + 446298*x^5 +...
A(x)^6 = 1 + 18*x + 261*x^2 + 3624*x^3 + 50076*x^4 + 695934*x^5 +...
where A(x) = 1 + x*A(x) + x*A(x)^2 + x*(1+x)*A(x)^3 + x^2*A(x)^4 + x^2*A(x)^5 + x^3*A(x)^6.
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x*A^2)*(1+x*A^3)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 29 2012
STATUS
approved