A243156 G.f. satisfies: x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^3) such that A(0) = 1.
1, 1, 3, 9, 28, 92, 319, 1154, 4302, 16382, 63391, 248499, 984867, 3940121, 15891386, 64545971, 263783729, 1083883910, 4475194635, 18557356409, 77251869363, 322723617687, 1352518263334, 5684939482522, 23959266771808, 101226312702475, 428650606083144, 1818991203750774, 7734098181837847
Offset: 0
Keywords
Examples
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)).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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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 *) -
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
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
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