A243632 Expansion of ((1-sqrt(1-4*x))/2-sqrt(((-sqrt(1-4*x)-11)*(1-sqrt(1-4*x)))/4+1)+1)/4.

0, 1, 2, 7, 30, 144, 744, 4047, 22858, 132830, 789124, 4771086, 29259992, 181569062, 1137891460, 7191411375, 45780189690, 293282202470, 1889328747180, 12231207808050, 79532035376500, 519196901292440

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/n)*Sum_{k=0..n} binomial(n+k-1,k)*binomial(3*n-2,n-4*k-1), n>1, a(0)=0.

A(x) = x*C(x)*S(x*C(x)), where C(x) is g.f. of A000108, S(x) is g.f. of A001003.

a(n) ~ (5*sqrt(2)+7)^(n-1/2) / (sqrt(4*sqrt(2)-5) * sqrt(Pi) * n^(3/2) * 2^(n+5/4)). - Vaclav Kotesovec, Jun 08 2014

O.g.f. A(x) is the series reversion of x*((1 - x)^4 - x^4)/(1 - x)^2. x*A'(x)/A(x) is the o.g.f. for A243644. - Peter Bala, Oct 02 2015

D-finite with recurrence 20*n*(n-1)*(n-2)*a(n) -2*(n-1)*(n-2)*(148*n-297)*a(n-1) +(n-2)*(1375*n^2 -7078*n +9159)*a(n-2) +(-1913*n^3 +17718*n^2 -54139*n +54510)*a(n-3) -4*(2*n-9)*(65*n^2 -561*n +1216) *a(n-4) -4*(n-5) *(2*n-9) *(2*n-11) *a(n-5)=0. - R. J. Mathar, Jul 20 2023

Mathematica

CoefficientList[Series[(3 - Sqrt[1-4*x] - Sqrt[10*Sqrt[1-4*x] - 4*x - 6])/8, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 08 2014 *)

Prog

(Maxima) a(n):=sum(binomial(n+k-1, k)*binomial(3*n-2, n-4*k-1), k, 0, n)/n;
(PARI) a(n) = if(n==0, 0, sum(k=0, n, binomial(n+k-1, k)*binomial(3*n-2, n-4*k-1)) / n); \\ Altug Alkan, Oct 02 2015

Crossrefs

Cf. A000108, A001003, A243644.

Sequence in context: A366055 A368932 A369160 * A196148 A193464 A166990

Adjacent sequences: A243629 A243630 A243631 * A243633 A243634 A243635

Keyword

nonn

Author

Vladimir Kruchinin, Jun 08 2014

Status

approved