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A093119 Number of convex polyominoes with a 3 X n+1 minimal bounding rectangle.

%I A093119 #21 Feb 01 2025 19:53:23
%S A093119 13,68,222,555,1171,2198,3788,6117,9385,13816,19658,27183,36687,48490,
%T A093119 62936,80393,101253,125932,154870,188531,227403,271998,322852,380525,
%U A093119 445601,518688,600418,691447,792455,904146,1027248,1162513
%N A093119 Number of convex polyominoes with a 3 X n+1 minimal bounding rectangle.
%H A093119 Colin Barker, <a href="/A093119/b093119.txt">Table of n, a(n) for n = 1..1000</a>
%H A093119 V. J. W. Guo and J. Zeng, <a href="https://arxiv.org/abs/math/0403262">The number of convex polyominoes and the generating function of Jacobi polynomials</a>, arXiv:math/0403262 [math.CO], 2004.
%H A093119 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A093119 a(n) = ((3*n+2)*C(2n+4, 4) - 4*n*C(n+2, n)^2)/(n+2), n>0.
%F A093119 a(n) = (6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6.
%F A093119 From _Colin Barker_, Feb 24 2019: (Start)
%F A093119 G.f.: x*(13 + 3*x + 12*x^2 - 5*x^3 + x^4) / (1 - x)^5.
%F A093119 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
%F A093119 (End)
%F A093119 E.g.f.: -1 + (6 + 72*x + 129*x^2 + 56*x^3 + 6*x^4)*exp(x)/6. - _G. C. Greubel_, Jun 26 2019
%t A093119 a[n_] := n^4 + 10*n^3/3 + 9*n^2/2 + 19*n/6 + 1; Array[a, 40] (* _Jean-François Alcover_, Feb 24 2019 *)
%o A093119 (PARI) Vec(x*(13 + 3*x + 12*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^40)) \\ _Colin Barker_, Feb 24 2019
%o A093119 (Magma) [(6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6: n in [1..40]]; // _G. C. Greubel_, Jun 26 2019
%o A093119 (Sage) [(6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6 for n in (1..40)] # _G. C. Greubel_, Jun 26 2019
%o A093119 (GAP) List([1..40], n-> (6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6); # _G. C. Greubel_, Jun 26 2019
%Y A093119 Row 2 of triangle A093118.
%K A093119 nonn,easy
%O A093119 1,1
%A A093119 _Ralf Stephan_, Mar 21 2004