OFFSET
1,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..550
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
FORMULA
G.f. A(x)=y satisfies (x-x^2-x^3)+y*(-1+3*x-x^3)+y^2*(-1+x^2-x^3)+y^3*(1-x+x^2-x^3)=0.
a(n) ~ sqrt((-1 - 3*s + s^3 - 2*r*(-1 + s^2 + s^3) + 3*r^2*(1 + s + s^2 + s^3)) / (1 - 3*s + 3*r*s - r^2*(1 + 3*s) + r^3*(1 + 3*s))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 0.3637032853572807454... and s = 0.8232781794881572707... are roots of the system of equations 3*r = 1 + r^3 + 2*(1 - r^2 + r^3)*s + 3*(-1 + r - r^2 + r^3)*s^2, s + s^2 + r*(-1 - 3*s + s^3) + r^3*(1 + s + s^2 + s^3) = s^3 + r^2*(-1 + s^2 + s^3). - Vaclav Kotesovec, Mar 10 2014
MATHEMATICA
terms = 32; A[_] = 0;
Do[A[x_] = (x-x^2-x^3) + (3x-x^3) A[x] + (-1+x^2-x^3) A[x]^2 + (1-x+x^2-x^3) A[x]^3 + O[x]^(terms+1) // Normal, {terms+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Nov 10 2018 *)
PROG
(PARI) a(n)=local(A); if(n<1, 0, A=O(x); for(k=1, n, A=(x-x^2-x^3)+A*(3*x-x^3)+A^2*(-1+x^2-x^3)+A^3*(1-x+x^2-x^3)); polcoeff(A, n))
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 10 2003
STATUS
approved