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%I #26 Jun 10 2022 06:14:24
%S 1,3,15,75,371,1833,9057,44753,221137,1092699,5399327,26679563,
%T 131831075,651413681,3218814561,15905050017,78591236385,388340962771,
%U 1918899743823,9481812581835,46852249642771
%N The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.
%C Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.
%C Related to A002378 by an Invert Transform.
%H Mudit Aggarwal and Samrith Ram, <a href="https://arxiv.org/abs/2206.04437">Generating functions for straight polyomino tilings of narrow rectangles</a>, arXiv:2206.04437 [math.CO], 2022.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 34.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1406.7788">Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices</a>, arXiv:1406.7788 [math.CO], eq. (24).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,4,-1).
%F G.f.: (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4).
%F a(n) = Sum_{k = 0..n} binomial(n + 3*k, 4*k)*2^k = Sum_{k = 0..n} A109960(n,k)*2^k. - _Peter Bala_, Nov 02 2017
%F a(n) = hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128). - _Peter Luschny_, Nov 02 2017
%p g := (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4) ;
%p taylor(%,x=0,30) ; gfun[seriestolist](%) ;
%p # Alternatively:
%p a := n -> hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128):
%p seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Nov 02 2017
%t LinearRecurrence[{6, -6, 4, -1}, {1, 3, 15, 75}, 21] (* _Jean-François Alcover_, Jul 14 2018 *)
%Y Cf. A003269 (4Xn floor), A236580 - A236582, A109960.
%K nonn,easy
%O 0,2
%A _R. J. Mathar_, Jan 29 2014
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