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%I #57 Jan 30 2020 21:29:15
%S 1,2,12,56,304,1632,9024,50304,283392,1607168,9167872,52537344,
%T 302239744,1744412672,10096263168,58576306176,340566147072,
%U 1983765676032,11574393962496,67631502065664,395710949228544,2318088492023808,13594307705438208,79802741538422784,468895276304695296
%N Expansion of 1/sqrt(1 - 4*x - 12*x^2).
%C Central coefficient of (1 + 2x + 4x^2)^n.
%C 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
%C 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
%H Seiichi Manyama, <a href="/A098453/b098453.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%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 G.f.: 1/sqrt((1+2*x)*(1-6*x)).
%F E.g.f.: exp(2*x)*BesselI(0, 4*x).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2*(n-k), n)*3^k.
%F 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
%F a(n) ~ sqrt(3)*6^n/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%F 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
%F a(n) = (-2)^n*hypergeom([-n,1/2], [1], 4). - _Peter Luschny_, Apr 26 2016
%F a(n) = 2^n*GegenbauerC(n, -n, -1/2)). - _Peter Luschny_, May 08 2016
%p seq(simplify((-2)^n*hypergeom([-n,1/2], [1], 4)),n=0..20); # _Peter Luschny_, Apr 26 2016
%p T := proc(n, k) option remember;
%p if n < 0 or k < 0 then 0
%p elif n = 0 then binomial(2*k, k)
%p else 2*(T(n-1, k+1) - T(n-1, k)) fi end:
%p a := n -> T(n, 0): seq(a(n), n=0..20); # _Peter Luschny_, Aug 23 2017
%t Table[SeriesCoefficient[1/Sqrt[1-4*x-12*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%o (PARI) x='x+O('x^66); Vec(1/sqrt(1-4*x-12*x^2)) \\ _Joerg Arndt_, May 11 2013
%Y Cf. A000984, A084609, A084770.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 08 2004
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