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%I #46 Jul 13 2019 10:10:02
%S 1,4,20,112,664,4064,25376,160640,1027168,6618496,42904960,279503360,
%T 1828222720,11999226880,78984381440,521218322432,3447059138048,
%U 22840932997120,151607254267904,1007830488424448,6708862677274624
%N Binomial transform of central Delannoy numbers A001850.
%C The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005
%C Coefficient of x^n in (1 + 4*x + 2*x^2)^n - _N-E. Fahssi_, Jan 17 2008
%C Number of paths from (0,0) to (n,0) using only steps U=(1,1), H=(1,0) and D=(1,-1), U can have 2 colors and H can have 4 colors. - _N-E. Fahssi_, Jan 27 2008
%H Vincenzo Librandi, <a href="/A080609/b080609.txt">Table of n, a(n) for n = 0..1000</a>
%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 - 8*x + 8*x^2 ).
%F a(n) = Sum_{k=0..n} binomial(n,k) * A001850(k).
%F E.g.f.: exp(4*x)*BesselI(0, 2*sqrt(2)*x). - _Vladeta Jovovic_, Mar 21 2004
%F Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/sqrt(2*Pi*n). - _Vaclav Kotesovec_, Oct 13 2012
%F G.f.: G(0), where G(k)= 1 + 4*x*(1-x)*(4*k+1)/(2*k+1 - 2*x*(1-x)*(2*k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2013
%F a(n) = LegendreP_n(sqrt(2))*8^(n/2). - _Vladimir Reshetnikov_, Nov 01 2015
%t Table[SeriesCoefficient[Series[1/Sqrt[1-8x+8x^2], {x, 0, n}], n], {n, 0, 12}]
%t Table[LegendreP[n, Sqrt[2]] 8^(n/2), {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%o (PARI) x='x+O('x^66); Vec(1/sqrt(1-8*x+8*x^2)) \\ _Joerg Arndt_, May 07 2013
%K easy,nonn
%O 0,2
%A _Emanuele Munarini_, Feb 26 2003
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