login
Number of convex polyominoes with an n+1 X n+1 minimal bounding square.
2

%I #16 Sep 08 2022 08:45:13

%S 5,68,1110,19010,329126,5693968,98074332,1680306750,28638104550,

%T 485731377560,8202665891316,137983709713668,2313146752430780,

%U 38658742468243520,644327268137066104,10712855104602481206

%N Number of convex polyominoes with an n+1 X n+1 minimal bounding square.

%H G. C. Greubel, <a href="/A093120/b093120.txt">Table of n, a(n) for n = 1..825</a>

%H 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.

%F a(n) = ((2+n)*C(4*n, 2*n) - 2*n*C(2*n, n)^2)/2, n>0.

%t a[n_]:= 1/2 (n+2) Binomial[4n, 2n] - n Binomial[2n, n]^2;

%t Array[a, 16] (* _Jean-François Alcover_, Jan 15 2019 *)

%o (PARI) a(n) = ((2+n)*binomial(4*n, 2*n) - 2*n*binomial(2*n, n)^2)/2; \\ _G. C. Greubel_, Jun 26 2019

%o (Magma) [((2+n)*Binomial(4*n, 2*n) - 2*n*Binomial(2*n, n)^2)/2: n in [1..20]]; // _G. C. Greubel_, Jun 26 2019

%o (Sage) [((2+n)*binomial(4*n, 2*n) - 2*n*binomial(2*n, n)^2)/2 for n in (1..20)] # _G. C. Greubel_, Jun 26 2019

%o (GAP) List([1..20], n-> ((2+n)*Binomial(4*n, 2*n) - 2*n*Binomial(2*n, n)^2)/2) # _G. C. Greubel_, Jun 26 2019

%Y Main diagonal of triangle A093118.

%K nonn

%O 1,1

%A _Ralf Stephan_, Mar 21 2004