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A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n. 7
1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008

LINKS

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

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

FORMULA

E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003

a(n) is also the central coefficient of (4+x+x^2)^n; a(n)=sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907 - N-E. Fahssi, Mar 30 2008

a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). [Peter Luschny, Sep 12 2011]

a(n+2)=( (2*n+3)*a(n+1) + 15*(n+1)*a(n) )/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012

a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = 2^n*GegenbauerC(n, -n, -1/4). - Peter Luschny, May 08 2016

MAPLE

a := n -> simplify(2^n*GegenbauerC(n, -n, -1/4)):

seq(a(n), n=0..22); # Peter Luschny, May 08 2016

MATHEMATICA

Table[n!*SeriesCoefficient[E^x*BesselI[0, 4*x], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PROG

(PARI) for(n=0, 30, t=polcoeff((1+x+4*x^2)^n, n, x); print1(t", "))

CROSSREFS

Cf. A002426, A084600-A084604, A084606-A084615.

Sequence in context: A227078 A146365 A146373 * A098773 A089998 A014728

Adjacent sequences:  A084602 A084603 A084604 * A084606 A084607 A084608

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 01 2003

STATUS

approved

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Last modified December 2 12:56 EST 2016. Contains 278678 sequences.