|
|
A091148
|
|
Expansion of (1-x-sqrt(1-2x-19x^2))/(10x^2).
|
|
4
|
|
|
1, 1, 6, 16, 81, 301, 1451, 6231, 29891, 137731, 666976, 3193026, 15658831, 76719891, 380788006, 1894818776, 9502977851, 47822585931, 241944876266, 1228151169656, 6258922649451, 31992657321551, 164040821525031
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 5 colors (i.e. Motzkin paths with the up steps in 5 colors). Series reversion of x/(1+x+5x^2). - Paul Barry, May 16 2005
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 2/(1-x+sqrt(1-2x-19x^2)).
a(n) = sum{k=0..n, binomial(n, k)5^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n).
a(n) = sum{k=0..n, C(n, 2k)C(k)5^k}; - Paul Barry, May 16 2005
D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
a(n) ~ 1/10*sqrt(230+61*sqrt(5))/(n^(3/2)*sqrt(Pi))*(1+2*sqrt(5))^n. - Vaclav Kotesovec, Sep 29 2012
G.f.: 1/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017
|
|
MATHEMATICA
|
CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 19 x^2]) / (10 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2013 *)
|
|
PROG
|
(PARI) x='x+O('x^66); Vec((1-x-sqrt(1-2*x-19*x^2))/(10*x^2)) \\ Joerg Arndt, May 11 2013
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|