A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2x^2)^n.

1, 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693, 243954113, 906758785, 3375949829, 12587460557, 46995614449, 175669746209, 657370655045, 2462383495357, 9232029156001

Offset

0,3

Comments

The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U (or D) can have 2 colors. - N-E. Fahssi, Feb 05 2008

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1718 (terms 0..120 from Vincenzo Librandi)

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, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004

a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(n, k)2^k}. - Paul Barry, Aug 26 2004

Sum[k=0..n, Trinomial(k, n) Binomial(n, k) ], with Trinomial=A027907. - Ralf Stephan, Jan 28 2005

a(n) is also the central coefficient of (2+x+x^2)^n; a(n)=sum_{k=0..n} 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, Feb 05 2008

a(n+2)=( (2*n+3)*a(n+1) + 7*(n+1)*a(n) )/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012

G.f.: G(0), where G(k)= 1 + x*(2+7*x)*(4*k+1)/( 4*k+2 - x*(2+7*x)*(4*k+2)*(4*k+3)/(x*(2+7*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013

a(n) ~ sqrt(8+2*sqrt(2)) * (1+2*sqrt(2))^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 09 2014

a(n) = hypergeom([1/2 - n/2, -n/2], [1], 8). - Peter Luschny, Mar 18 2018

Maple

a := n -> hypergeom([1/2 - n/2, -n/2], [1], 8):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Mar 18 2018

Mathematica

CoefficientList[Series[1/Sqrt[1-2x-7x^2], {x, 0, 30}], x] (* Harvey P. Dale, Sep 18 2011 *)

Prog

(PARI) for(n=0, 30, t=polcoeff((1+x+2*x^2)^n, n, x); print1(t", "))
(Maxima) a(n):=coeff(expand((1+x+2*x^2)^n), x, n);
makelist(a(n), n, 0, 12); /* Emanuele Munarini, Mar 02 2011 */

Crossrefs

Cf. A002426, A084600, A084602-A084615.

Sequence in context: A220900 A138277 A343549 * A149532 A149533 A149534

Adjacent sequences: A084598 A084599 A084600 * A084602 A084603 A084604

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2003

Status

approved