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
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.
Programs
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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
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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 *)
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Maxima
a(n):=coeff(expand((1+2*x+3*x^2)^n),x,n); makelist(a(n),n,0,12);
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PARI
for(n=0,30,t=polcoeff((1+2*x+3*x^2)^n,n,x); print1(t","))
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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
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
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