OFFSET
2,2
COMMENTS
Number of lattice paths from (0,0) to (2n,4), using steps U=(1,1), D=(1,-1) and at even levels(zero, positive and negative) also H=(2,0). Example: a(3)=9 because we have UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, HUUUU, UUHUU and UUUUH. - Emeric Deutsch, Jan 30 2004
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..1000
FORMULA
From Emeric Deutsch, Jan 30 2004: (Start)
a(n) = [t^(n+2)](1+3t+t^2)^n.
a(n) = Sum_{j=ceiling((n+2)/2)..n} (3^(2j-n-2)*binomial(n, j)*binomial(j, n+2-j)). (End)
From Paul Barry, Sep 20 2004: (Start)
E.g.f.: exp(3x) * BesselI(2, 2x);
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+2). (End)
Conjecture: n*(n+4)*a(n) - 3*(n+2)*(2*n+3)*a(n-1) + 5*(n+2)*(n+1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
G.f.: (3*x - 1 + (1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2). - Mark van Hoeij, Apr 18 2013
a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2013
Assuming offset 0: a(n) = C(2*n+4,n)*hypergeom([-n,-n-4],[-3/2-n],-1/4). - Peter Luschny, May 09 2016
MAPLE
series( (3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2), x=0, 30); # Mark van Hoeij, Apr 18 2013
MATHEMATICA
CoefficientList[Series[(3*x-1+(1-6*x+7*x^2)/Sqrt[5*x^2-6*x+1])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2013 *)
PROG
(PARI) x='x+O('x^66); Vec((3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2)) /* Joerg Arndt, Apr 19 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved