A107265 Expansion of (1-5*x-sqrt((1-5*x)^2-4*5*x^2))/(2*5*x^2).

1, 5, 30, 200, 1425, 10625, 81875, 646875, 5211875, 42659375, 353725000, 2965031250, 25083859375, 213894609375, 1836516718750, 15863968750000, 137767560546875, 1202116083984375, 10534061644531250, 92664360625000000, 817975366904296875, 7243402948779296875

Offset

0,2

Comments

Series reversion of x/(1+5x+5x^2). Transform of 5^n under the matrix A107131. A row of A107267.

Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 5 colors and D(1,-1) one color. - Paul Barry, May 16 2005

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2).

a(n) = Sum_{k=0..n} (1/(k+1)) * C(k+1,n-k+1) * C(n, k) * 5^k.

E.g.f.: a(n) = n!* [x^n] exp(5*x)*BesselI(1,2*sqrt(5)*x) /(sqrt(5)*x). -Peter Luschny, Aug 25 2012

D-finite with recurrence: (n+2)*a(n) = 5*(2*n+1)*a(n-1) - 5*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(38+17*sqrt(5))*(5+2*sqrt(5))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

G.f.: 1/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

Mathematica

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+5*x^2])/(10*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Prog

(PARI) x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2)) \\ Joerg Arndt, May 15 2013

Crossrefs

Sequence in context: A103433 A081015 A090139 * A196678 A128328 A245376

Adjacent sequences: A107262 A107263 A107264 * A107266 A107267 A107268

Keyword

easy,nonn

Author

Paul Barry, May 15 2005

Status

approved