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A059682 Triangle T(n,k) giving number of 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).

%I A059682 #17 May 12 2019 09:00:56
%S A059682 1,0,1,0,1,3,0,0,2,3,0,0,1,6,5,0,0,0,2,11,5,0,0,0,1,10,19,7,0,0,0,0,3,
%T A059682 22,28,7,0,0,0,0,1,15,52,40,9,0,0,0,0,0,3,45,90,53,9,0,0,0,0,0,1,21,
%U A059682 119,158,69,11,0,0,0,0,0,0,4,73,257,238,86,11,0,0,0,0,0,0,1,28,237,505,360,106,13
%N A059682 Triangle T(n,k) giving number of 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).
%H A059682 R. C. Read, <a href="http://cms.math.ca/cjm/v14/cjm1962v14.0001-0020.pdf">Contributions to the cell growth problem</a>, Canad. J. Math., 14 (1962), 1-20. See Eq. (8.5) for g.f..
%F A059682 G.f.: ((y^9-y^8)*x^6+y^8*x^5+(y^7-2*y^6+y^5)*x^4+(y^6-y^3)*x^3+(-y^4+y^2)*x^2+(-y^2-y)*x+1)*y^2*x/((y^3*x^2+(y^2+y)*x-1)*(y*x-1)*(y^2*x-1)*(y^6*x^4+(y^4+y^2)*x^2-1)). - _Vladeta Jovovic_, Apr 02 2002
%e A059682 Triangle starts:
%e A059682 1;
%e A059682 0,1;
%e A059682 0,1,3;
%e A059682 0,0,2,3;
%e A059682 ...
%t A059682 rows = 13; gf = ((y^9 - y^8)*x^6 + y^8*x^5 + (y^7 - 2*y^6 + y^5)*x^4 + (y^6 - y^3)*x^3 + (-y^4 + y^2)*x^2 + (-y^2 - y)*x + 1)*y^2*x/((y^3*x^2 + (y^2 + y)*x - 1)*(y*x - 1)*(y^2*x - 1)*(y^6*x^4 + (y^4 + y^2)*x^2 - 1));
%t A059682 coes = CoefficientList[#, x]& /@ CoefficientList[gf + O[y]^(rows+2), y];
%t A059682 T[n_, k_] := coes[[n+1, k+1]];
%t A059682 Table[T[n, k], {n, 2, rows+1}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Nov 12 2017, after _Vladeta Jovovic_ *)
%Y A059682 Main diagonal gives A109613(n-2).
%Y A059682 Cf. A059683 (3xk), A059684 (4xk).
%K A059682 nonn,easy,nice,tabl
%O A059682 2,6
%A A059682 _N. J. A. Sloane_, Feb 05 2001
%E A059682 More terms from _Vladeta Jovovic_, Apr 02 2002