0,2

More generally, if G(x) satisfies

G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then

G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^2) / (1+x)^2 ).

Recurrence: 31*(n-1)*n*(n+1)*(5974*n^3 - 40359*n^2 + 90115*n - 67124)*a(n) = 2*(n-1)*n*(1003632*n^4 - 7282128*n^3 + 18518502*n^2 - 18822839*n + 5649607)*a(n-1) - 2*(n-1)*(740776*n^5 - 6486068*n^4 + 21715762*n^3 - 34616651*n^2 + 26123385*n - 7413210)*a(n-2) + 2*(2*n - 5)*(65714*n^5 - 575377*n^4 + 1957337*n^3 - 3264653*n^2 + 2726129*n - 941430)*a(n-3) + 4*(n-3)*(2*n - 7)*(2*n - 5)*(5974*n^3 - 22437*n^2 + 27319*n - 11394)*a(n-4). - Vaclav Kotesovec, Nov 18 2017

a(n) ~ sqrt((s*(1+r*s)*(2 + s + 3*r*s^2)) / (1 + r*(1 + 6*s*(1+r*s)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.099424837262345547872398211374352678... and s = 2.183663565361369673488934371066403742... are roots of the system of equations (1 + r*s)^2*(1 + r*s^2) = s, 2*r*(1 + s + 2*r^2*s^3 + r*s*(1 + 3*s)) = 1. - Vaclav Kotesovec, Nov 18 2017

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+2*k+2,n-k). - Seiichi Manyama, Jan 27 2024

G.f.: A(x) = 1 + 3*x + 15*x^2 + 94*x^3 + 661*x^4 + 4983*x^5 + 39363*x^6 + 321587*x^7 +...

(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^2 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

Cf. A181734, A274735.

Sequence in context: A369301 A379081 A368964 * A378951 A177341 A220262

Adjacent sequences: A274731 A274732 A274733 * A274735 A274736 A274737

Paul D. Hanna, Aug 02 2016

approved