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%I #18 Feb 01 2023 16:42:15
%S 1,16,96,400,1408,4480,13312,37632,102400,270336,696320,1757184,
%T 4358144,10649600,25690112,61276160,144703488,338690048,786432000,
%U 1812987904,4152360960,9453961216,21407727616,48234496000,108179488768,241591910400,537407782912
%N Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).
%C An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by the g.f. 2*x*y/(1-2*(x+y-x*y)).
%H G. C. Greubel, <a href="/A100313/b100313.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F G.f.: 1 + 16*x*(1-x)^2/(1-2*x)^4.
%F a(n) = (1/3) n*(n^2 + 9*n + 14) * 2^n for n>0, with a(0) = 1.
%F a(n) = 16 * A055585(n-1) for n>0.
%F E.g.f.: (1/3)*(3 + 8*x*(6 + 6*x + x^2)*exp(2*x)). - _G. C. Greubel_, Feb 01 2023
%t Table[If[n==0, 1, 2^n*n*(n^2+9*n+14)/3], {n,0,40}] (* _G. C. Greubel_, Feb 01 2023 *)
%o (PARI) vector(50, n, n*(n^2+9*n+14) * 2^n / 3) \\ _Michel Marcus_, Dec 01 2014
%o (Magma) [2^n*n*(n^2+9*n+14)/3 +0^n: n in [0..40]]; // _G. C. Greubel_, Feb 01 2023
%o (SageMath) [2^n*n*(n^2+9*n+14)/3 +0^n for n in range(41)] # _G. C. Greubel_, Feb 01 2023
%Y Cf. A055585, A100312 (m=3), this sequence (m=4).
%K nonn,easy
%O 0,2
%A _Sergey Kitaev_, Nov 13 2004
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 21 2018
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