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%I #66 Mar 27 2023 03:44:15
%S 1,2,10,44,214,1052,5284,26840,137638,710828,3692140,19266920,
%T 100932220,530479640,2795917960,14771797424,78210099718,414862155980,
%U 2204273582236,11729283976136,62496686731924,333400654676168
%N 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.
%C 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
%C Self-convolution of a(n)/2^n gives A002605(n+1). - _Vladimir Reshetnikov_, Oct 10 2016
%C 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
%H Seiichi Manyama, <a href="/A084609/b084609.txt">Table of n, a(n) for n = 0..1358</a> (terms 0..200 from Vincenzo Librandi)
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F a(n) = Sum_{k = 0..floor(n/2)} C(n,k)*C(2*(n-k),n)*2^k. - _Paul Barry_, Sep 08 2004
%F 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
%F E.g.f.: exp(2*x) * Bessel_I(0,2*sqrt(3)*x)
%F 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
%F a(n) ~ sqrt(18+6*sqrt(3))*(2+2*sqrt(3))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F 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
%F 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
%F 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
%F 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
%t (* Programs from _Robert G. Wilson v_, Mar 02 2011 *)
%t a[n_]:= Sum[Binomial[n, k] Binomial[2(n-k), n] 2^k, {k, 0, n/2}]; Array[a, 30, 0]
%t a[n_]:= CoefficientList[Expand[(1 +2x +3x^2)^n], x][[n+1]]; Array[a, 30, 0]
%t CoefficientList[Series[1/Sqrt[1 -4x -8x^2], {x,0,30}], x]
%t Range[0, 30]! CoefficientList[ Series[ Exp[ 2x] BesselI[0, Sqrt[12] x], {x, 0, 30}], x] (* End *)
%t Table[2^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 3], {n,0,30}] (* _Vladimir Reshetnikov_, Oct 10 2016 *)
%o (PARI) for(n=0,30,t=polcoeff((1+2*x+3*x^2)^n,n,x); print1(t","))
%o (Maxima) a(n):=coeff(expand((1+2*x+3*x^2)^n),x,n);
%o makelist(a(n),n,0,12);
%o (Magma)
%o A084609:= func< n | (&+[Binomial(n,j)*Binomial(2*(n-j),n)*2^j: j in [0..Floor(n/2)]]) >;
%o [A084609(n): n in [0..50]]; // _G. C. Greubel_, Mar 26 2023
%o (SageMath)
%o def A084609(n): return sum(binomial(n,j)*binomial(2*(n-j),n)*2^j for j in range(n//2+1))
%o [A084609(n) for n in range(51)] # _G. C. Greubel_, Mar 26 2023
%Y Cf. A002426, A084600 - A084608, A084610 - A084615.
%Y Row sums of A328347.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Jun 01 2003
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