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%I #21 Sep 08 2022 08:45:01
%S 0,0,0,0,0,0,0,0,1,12,92,576,3214,16664,82160,390656,1807781,8192524,
%T 36519556,160645504,699030226,3014470024,12901501696,54863119744,
%U 232022899306,976598630968,4093581923320,17096805375360,71176501409756
%N Number of 1-punctured staircase polygons (by perimeter) with a hole of perimeter 4.
%H Vincenzo Librandi, <a href="/A055022/b055022.txt">Table of n, a(n) for n = 0..500</a>
%H A. J. Guttmann et al., <a href="https://arxiv.org/abs/cond-mat/0003441">Punctured polygons and polyominoes on the square lattice</a>, arXiv:cond-mat/0003441 [cond-mat.stat-mech], 2000.
%H A. J. Guttmann et al., <a href="https://doi.org/10.1088/0305-4470/33/9/303">Punctured polygons and polyominoes on the square lattice</a>, J. Physics A: Math. and Gen, 33 (9) (2000), 1735-1764.
%F D-finite with recurrence n*(n-8)*a(n) +2*(-4*n^2+35*n-45)*a(n-1) +8*(2*n-9)*(n-5)*a(n-2)=0. - _R. J. Mathar_, Aug 14 2012
%F For n>3, a(n) = 4^(n-4)-binomial(2n,n)(n-3)(n^2-5n+10)/(4(2n-1)(2n-3)(2n-5)). - _Michael D. Weiner_, Jan 17 2018
%p gf := (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)/(2*(1 - 4*x)) - (1 - 6*x + 10*x^2 - 4*x^3)/(2*sqrt(1 - 4*x)):
%p s := series(gf, x, 50):
%p for i from 0 to 50 do printf(`%d,`,coeff(s,x,i)) od:
%t Join[{0, 0, 0, 0}, Table[4^(n - 4) - Binomial[2 n, n] (n - 3) (n^2 - 5 n + 10) / (4 (2 n - 1) (2 n - 3) (2 n - 5)), {n, 4, 50}]] (* _Vincenzo Librandi_, Jan 20 2018 *)
%o (Magma) [0,0,0,0] cat [4^(n-4)-Binomial(2*n,n)*(n-3)*(n^2-5*n+10) div (4*(2*n-1)*(2*n-3)*(2*n-5)): n in [4..30]]; // _Vincenzo Librandi_, Jan 20 2018
%Y Cf. A055024 (hole perimeter 6).
%K easy,nonn
%O 0,10
%A _James A. Sellers_, May 31 2000
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