A084609 Coefficients of 1/sqrt(1-4*x-8*x^2); also, a(n) is the central coefficient of (1+2*x+3*x^2)^n.

1, 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424, 78210099718, 414862155980, 2204273582236, 11729283976136, 62496686731924, 333400654676168

Offset

0,2

Comments

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U can have 3 colors and H can have 2 colors. - N-E. Fahssi, Mar 30 2008

Self-convolution of a(n)/2^n gives A002605(n+1). - Vladimir Reshetnikov, Oct 10 2016

The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1358 (terms 0..200 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

a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(2*(n-k),n)*2^k. - Paul Barry, Sep 08 2004

a(n) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*3^k*2^(n-2*k); a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(n-k,k)*3^k*2^(n-2k). - Paul Barry, Sep 19 2006

E.g.f.: exp(2*x) * Bessel_I(0,2*sqrt(3)*x)

a(n) = ( 2*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2) )/n, a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 20 2012

a(n) ~ sqrt(18+6*sqrt(3))*(2+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - 2*x*(1+2*x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k+1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013

G.f.: G(0), where G(k)= 1 + x*(2+4*x)*(4*k+1)/(2*k+1 - x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(1+2*x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 18 2013

a(n) = 2^n * hypergeom([(1-n)/2,-n/2], [1], 3) = binomial(2*n, n) * hypergeom([(1-n)/2,-n/2], [1/2-n], -2). - Vladimir Reshetnikov, Oct 10 2016

a(n) = (-2*sqrt(-2))^n * P(n, sqrt(-1/2)), where P(n,x) denotes the n-th Legendre polynomial. - Peter Bala, Feb 07 2022

Mathematica

(* Programs from Robert G. Wilson v, Mar 02 2011 *)
a[n_]:= Sum[Binomial[n, k] Binomial[2(n-k), n] 2^k, {k, 0, n/2}]; Array[a, 30, 0]
a[n_]:= CoefficientList[Expand[(1 +2x +3x^2)^n], x][[n+1]]; Array[a, 30, 0]
CoefficientList[Series[1/Sqrt[1 -4x -8x^2], {x, 0, 30}], x]
Range[0, 30]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 30}], x] (* End *)
Table[2^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 3], {n, 0, 30}] (* Vladimir Reshetnikov, Oct 10 2016 *)

Prog

(PARI) for(n=0, 30, t=polcoeff((1+2*x+3*x^2)^n, n, x); print1(t", "))
(Maxima) a(n):=coeff(expand((1+2*x+3*x^2)^n), x, n);
makelist(a(n), n, 0, 12);
(Magma)
A084609:= func< n | (&+[Binomial(n, j)*Binomial(2*(n-j), n)*2^j: j in [0..Floor(n/2)]]) >;
[A084609(n): n in [0..50]]; // G. C. Greubel, Mar 26 2023
(SageMath)
def A084609(n): return sum(binomial(n, j)*binomial(2*(n-j), n)*2^j for j in range(n//2+1))
[A084609(n) for n in range(51)] # G. C. Greubel, Mar 26 2023

Crossrefs

Cf. A002426, A084600 - A084608, A084610 - A084615.

Row sums of A328347.

Sequence in context: A100397 A084059 A339642 * A105485 A151313 A144896

Adjacent sequences: A084606 A084607 A084608 * A084610 A084611 A084612

Keyword

nonn,easy

Author

Paul D. Hanna, Jun 01 2003

Status

approved