A243626 Expansion of (1-3*x-sqrt(x^2-10*x+1))/(4*x+2).

0, 1, 4, 22, 142, 1006, 7570, 59410, 480910, 3986230, 33666154, 288675322, 2506487158, 21993277294, 194717676514, 1737297725602, 15604975886110, 140999418473830, 1280687414193370, 11686769594020810, 107093513405758342

Offset

0,3

Links

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

Formula

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

a(n) ~ sqrt(36+29*sqrt(6)) * (5+2*sqrt(6))^n / (50 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 08 2014

D-finite with recurrence: 2*n*a(n)-(30+19*n)*a(n+1)-(9+8*n)*a(n+2)+(n+3)*a(n+3)=0. - Robert Israel, Jan 07 2018

a(n) = Sum_{k=0..n-1} binomial(n-1,k)*hypergeom([k+1-n, n], [k+2], -2). - Peter Luschny, Jan 07 2018

O.g.f. A(x) = series reversion of x*(1 - x)/((1 + x)*(1 + 2*x)). - Peter Bala, Nov 08 2022

Maple

f:= gfun:-rectoproc({2*n*a(n)-(30+19*n)*a(n+1)-(9+8*n)*a(n+2)+(n+3)*a(n+3)=0, a(0)=0, a(1)=1, a(2)=4}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jan 07 2018

Mathematica

CoefficientList[Series[(1-3*x-Sqrt[x^2-10*x+1])/(4*x+2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 08 2014 *)

Prog

(Maxima)
a(n):=sum((k+1)*sum(2^i*binomial(n, n-k-i-1)*binomial(n+i-1, i), i, 0, n-k-1), k, 0, n-1)/n;
(PARI) x='x+O('x^50); concat([0], Vec((1-3*x - sqrt(x^2-10*x+1))/(4*x + 2))) \\ G. C. Greubel, Jun 02 2017

Crossrefs

Cf. A114710.

Sequence in context: A190271 A045744 A369504 * A369485 A104991 A368562

Adjacent sequences: A243623 A243624 A243625 * A243627 A243628 A243629

Keyword

nonn,easy

Author

Vladimir Kruchinin, Jun 08 2014

Status

approved