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%I #50 May 25 2023 07:29:26
%S 1,6,52,504,5136,53856,575296,6225792,68026624,748832256,8291791872,
%T 92255680512,1030537089024,11550176206848,129824329777152,
%U 1462841567576064,16518691986407424,186887008999047168,2117944490818011136,24038305911245635584,273199990096465494016
%N Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6*x+8*x^2)^n.
%C Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - _Paul Barry_, May 30 2005
%C 2^n*P_n(3), where P_n is the n-th Legendre polynomial. 2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 -4*k*x +4*x^2) and e.g.f. exp(2*k*x)*BesselI(0, 2*sqrt(k^2-1)*x). - _Paul Barry_, May 30 2005
%C Diagonal of rational functions 1/(1 - x - 2*y - 2*x*y), 1/(1 - x - 2*y*z - 2*x*y*z), 1/(1 - 2*x - y*z - 2*x*y*z). - _Gheorghe Coserea_, Jul 07 2018
%H Harvey P. Dale, <a href="/A084773/b084773.txt">Table of n, a(n) for n = 0..939</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and 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.
%F a(n) = 2*A052141(n) = 2^n * A001850(n), n>0.
%F From _Paul Barry_, May 30 2005: (Start)
%F E.g.f.: exp(6*x)*Bessel_I(0, 2*sqrt(8)*x).
%F a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2(n-k), n)*(-1)^k*3^(n-2*k). (End)
%F D-finite with recurrence: n*a(n) + 6*(-2*n+1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - x*(6-2*x)*(2*k+1)/(x*(6-2*x)*(2*k+1) + (k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013
%F a(n) = 2^n*hyper2F1(-n, -n, 1, 2). - _Peter Luschny_, May 20 2015
%F a(n) = A059474(n,n). - _Alois P. Heinz_, Oct 05 2017
%F a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Jul 11 2018
%e G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
%p a := n -> 2^n*hypergeom([-n, -n], [1], 2):
%p seq(simplify(a(n)), n=0..20); # _Peter Luschny_, May 20 2015
%t CoefficientList[Series[1/Sqrt[(1-12x+4x^2)],{x,0,20}],x] (* _Harvey P. Dale_, Dec 13 2017 *)
%t Table[2^n*LegendreP[n,3], {n,0,40}] (* _G. C. Greubel_, May 21 2023 *)
%o (PARI) for(n=0,30,t=polcoeff((1+6*x+8*x^2)^n,n,x); print1(t","))
%o (Magma) [2^n*Evaluate(LegendrePolynomial(n), 3): n in [0..40]]; // _G. C. Greubel_, May 21 2023
%o (SageMath) [2^n*gen_legendre_P(n,0,3) for n in range(41)] # _G. C. Greubel_, May 21 2023
%Y Sequences of the form 2^n*LegendreP(n, 2*m+1): A000079 (m=0), this sequence (m=1), A098270 (m=2).
%Y Cf. A001850, A052141, A059474.
%Y See A152254 for another interpretation.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 10 2003
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