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A084609 Coefficients of 1/(1-4x-8x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+3x^2)^n. 9
1, 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424, 78210099718, 414862155980, 2204273582236, 11729283976136, 62496686731924, 333400654676168 (list; graph; refs; listen; history; text; internal format)
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

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

MATHEMATICA

f[n_] := Sum[Binomial[n, k] Binomial[2 (n - k), n] 2^k, {k, 0, n/2}] (* Or *)

f[n_] := CoefficientList[ Expand[(1 + 2 x + 3 x^2)^n], x][[n + 1]]; Array[f, 22, 0] (* Or *)

CoefficientList[ Series[ 1/Sqrt[1 - 4 x - 8 x^2], {x, 0, 21}], x] (* Or *)

Range[0, 21]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 21}], x] (* Robert G. Wilson v *)

Table[2^n Hypergeometric2F1[(1 - n)/2, -n/2, 1, 3], {n, 0, 20}] (* 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);

CROSSREFS

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

Sequence in context: A099919 A100397 A084059 * A105485 A151313 A144896

Adjacent sequences:  A084606 A084607 A084608 * A084610 A084611 A084612

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 01 2003

STATUS

approved

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Last modified August 19 18:51 EDT 2019. Contains 326133 sequences. (Running on oeis4.)