A098453 Expansion of 1/sqrt(1 - 4*x - 12*x^2).

1, 2, 12, 56, 304, 1632, 9024, 50304, 283392, 1607168, 9167872, 52537344, 302239744, 1744412672, 10096263168, 58576306176, 340566147072, 1983765676032, 11574393962496, 67631502065664, 395710949228544, 2318088492023808, 13594307705438208, 79802741538422784, 468895276304695296

Offset: 0

Author

Paul Barry, Sep 08 2004

Keywords

• easy
• nonn

Comments

Central coefficient of (1 + 2x + 4x^2)^n.

a(n) is the number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 2 colors and the U steps can have 4 colors. - N-E. Fahssi, Mar 31 2008

a(n) is the number of 2 X n matrices with terms in {1,2,3}, same number of 1's in top and bottom rows, and no constant columns. For example, a(1)=2 counts the transposes of (2,3) and (3,2). The number of such matrices with k 1's in each row is binomial(n,2k) [choose columns containing 1's] * binomial(2k,k) [place 1's in these columns] * 2^n [place 2 or 3 in the topmost available spot in each column and the other of 2,3 in the other spot if not occupied by a 1]. - David Callan, Aug 25 2009

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Cf. A000984, A084609, A084770.

Programs

Maple

seq(simplify((-2)^n*hypergeom([-n,1/2], [1], 4)),n=0..20); # Peter Luschny, Apr 26 2016
T := proc(n, k) option remember;
if n < 0 or k < 0 then 0
elif n = 0 then binomial(2*k, k)
else 2*(T(n-1, k+1) - T(n-1, k)) fi end:
a := n -> T(n, 0): seq(a(n), n=0..20); # Peter Luschny, Aug 23 2017

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-4*x-12*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-4*x-12*x^2)) \\ Joerg Arndt, May 11 2013

Formula

G.f.: 1/sqrt((1+2*x)*(1-6*x)).

E.g.f.: exp(2*x)*BesselI(0, 4*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2*(n-k), n)*3^k.

D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) + 12*(n+1)*a(n))/(n+2); a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 30 2012

a(n) ~ sqrt(3)*6^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

G.f.: G(0), where G(k) = 1 + 2*x*(1+3*x)*(4*k+1)/( 2*k+1 - x*(1+3*x)*(2*k+1)*(4*k+3)/(x*(1+3*x)*(4*k+3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

a(n) = (-2)^n*hypergeom([-n,1/2], [1], 4). - Peter Luschny, Apr 26 2016

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