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A334551 Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.

%I A334551 #29 Apr 12 2024 09:28:48
%S A334551 1,4,25,120,497,1924,7265,27288,102745,388692,1477721,5643064,
%T A334551 21632785,83204260,320932177,1240939448,4808642313,18668848852,
%U A334551 72601081385,282762109272,1102772229313,4306062994148,16832791708257,65867445819160,257980829463017
%N A334551 Number of fixed polyominoes with 2n-1 cells and width and height both equal to n.
%C A334551 A polyomino with width and height equal to n must have at least 2n-1 cells.
%H A334551 Andrew Howroyd, <a href="/A334551/b334551.txt">Table of n, a(n) for n = 1..500</a>
%H A334551 Andrew Conway, <a href="https://doi.org/10.1088/0305-4470/28/2/011">Enumerating 2D percolation series by the finite-lattice method: theory</a>, J. Phys. A: Math. Gen., 28 (1995), 335-349. See Table 4.
%F A334551 a(n) = 2*binomial(2*(n-1), n-1) + 4*(n-2) + (n-2)^2*(2*n-5) + 2*Sum_{i=1..n-2} Sum_{j=1..n-2} ((n-2-i)*(n-2-j)+2)*binomial(i+j, i) for n > 1.
%F A334551 a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8. - _Peter J. Taylor_, Dec 15 2020
%F A334551 From _Stefano Spezia_, Sep 02 2022: (Start)
%F A334551 G.f.: 8*x/sqrt(1 - 4*x) - (8 - 17*x + 15*x^2)/(1 - x)^3.
%F A334551 a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)
%e A334551 a(3) = 25. Up to rotation and reflection there are 6 possibilities:
%e A334551        X       X   X         X         X     X
%e A334551        X     X X   X X X   X X X   X X X   X X X
%e A334551    X X X   X X     X       X       X         X
%t A334551 Array[8Binomial[2(#-1),#-1]-3#^2+4#-8&,50] (* _Paolo Xausa_, Dec 21 2023 *)
%o A334551 (PARI) a(n) = 8*binomial(2*(n-1), n-1) - 3*n^2 + 4*n - 8; \\ _Peter J. Taylor_, Dec 15 2020
%Y A334551 Main diagonal of A334552.
%Y A334551 Cf. A268404.
%K A334551 nonn
%O A334551 1,2
%A A334551 _Andrew Howroyd_, Jun 06 2020