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A243156 G.f. satisfies: x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^3) such that A(0) = 1. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

Cf. A243157, A115344.

Sequence in context: A176673 A215007 A105849 * A228449 A131203 A191637

Adjacent sequences:  A243153 A243154 A243155 * A243157 A243158 A243159

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 31 2014

STATUS

approved

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Last modified January 20 08:18 EST 2020. Contains 331081 sequences. (Running on oeis4.)