OFFSET
0,3
COMMENTS
Compare to A243157, the series reversion of x*(1 - x)/(1 - x - x^3).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f.: A(x) = x / Series_Reversion(x*(1 + Series_Reversion(x / (1 + 4*x + 3*x^2 + x^3)))).
G.f. satisfies: x = (1+x)*A(x) - A(x)^2 + x*A(x)^3 such that A(0) = 1.
a(n) ~ sqrt((s-1)*s/(3*r*s-1)) / (2*sqrt(Pi) * r^n * n^(3/2)), where r = 2/(3 + sqrt(13 + 16*sqrt(2))) = 0.22299351557517... and s = (1+sqrt(5+4*sqrt(2)))/2 = 2.1322418823119... . - Vaclav Kotesovec, May 31 2014
a(n) = floor(((6^(2*n)*(6^(-n)+1)^3+6^n)/(6*(1+6^n)))^n-6^n*floor((((6^(2*n)*(6^(-n)+1)^3+6^n)/(36*(1+6^n)))^n)))/n, for n>0. - Tani Akinari, May 20 2018
D-finite with recurrence n*(n+1)*a(n) -9*n*(n-1)*a(n-1) +6*(6*n^2 -22*n +17)*a(n-2) -27*(3*n-7) *(n-4)*a(n-3) +2*(14*n^2 -172*n +435)*a(n-4) +3*(53*n -192) *(n-5) *a(n-5) -341*(n-5)*(n-6) *a(n-6)=0. - R. J. Mathar, Jul 20 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 28*x^4 + 92*x^5 + 319*x^6 + 1154*x^7 + 4302*x^8 + 16382*x^9 + 63391*x^10 + ... where A(x) = x * (1 - A(x) - A(x)^3) / (1 - A(x)).
MATHEMATICA
CoefficientList[x/InverseSeries[x*(1+InverseSeries[Series[x/(1 + 4*x + 3*x^2 + x^3), {x, 0, 20}], x]), x], x] (* Vaclav Kotesovec, May 31 2014 after Paul D. Hanna *)
PROG
(PARI) {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/(1 + 4*x + 3*x^2 + x^3 +x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=0^n+floor(((6^(2*n)*(6^(-n)+1)^3+6^n)/(6*(1+6^n)))^n-6^n*floor((((6^(2*n)*(6^(-n)+1)^3+6^n)/(36*(1+6^n)))^n)))/(n+0^n)}
for(n=0, 50, print1(a(n), ", ")) \\ Tani Akinari, May 20 2018
(PARI) {a(n)=0^n+sum(k=1, n, binomial(n, k)*binomial(3*k-n, k-1))/(n+0^n)} \\ Tani Akinari, May 20 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2014
STATUS
approved