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A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

%I A360419 #15 Feb 23 2023 13:21:30
%S A360419 0,0,0,0,1,2,5,9,16,24,37,50,71,93,121,151,192,231,285,338,398,470,
%T A360419 548,626,723,827,924,1056,1175,1314,1454,1629,1763,1985,2138,2356,
%U A360419 2540,2820,2976,3305,3491,3834,4039,4441,4613,5103,5291,5775,5999,6572
%N A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.
%C A360419 A U-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLLLLLRR.
%H A360419 Andrew Howroyd, <a href="/A360419/b360419.txt">Table of n, a(n) for n = 1..1000</a>
%F A360419 G.f.: Sum_{k>=1} (x^k/(1 - x^k)) * (B(k+1, x)^2 + B(k+1, x^2))/2 where B(k, x) = Sum_{j>=k} x^j/(1 - x^j). - _Andrew Howroyd_, Feb 07 2023
%e A360419 a(5)=1 because of:
%e A360419   OO
%e A360419   O
%e A360419   OO
%e A360419 The a(7) = 5 polyominoes are:
%e A360419   O
%e A360419   O     O O             O
%e A360419   O O   O O   O OO   O  O   O   O
%e A360419   OOO   OOO   OOOO   OOOO   OOOOO
%o A360419 (PARI) B(n,k,x) = sum(j=k, n, x^j/(1 - x^j), O(x*x^n))
%o A360419 seq(n) = Vec(sum(k=1, (n-2)\3, x^k*(B(n-k, k+1, x)^2 + B((n-k)\2, k+1, x^2))/(1-x^k), O(x*x^n))/2, -n) \\ _Andrew Howroyd_, Feb 07 2023
%Y A360419 Cf. A028247, A270060, A360420, A360421.
%K A360419 nonn
%O A360419 1,6
%A A360419 _John Mason_, Feb 06 2023