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%I #10 Jul 26 2022 11:18:50
%S 0,1,1,3,9,27,117,459,2421,11979,74421,443979,3184821,22216779,
%T 180996021,1444706379,13186615221,118495279179,1198323664821,
%U 11969865775179,132880218064821,1460470704175179,17659740362704821
%N Number of deco polyominoes of height n, consisting only of columns of even length. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
%C a(n)=A121745(n,0).
%D E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
%D E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29- 42.
%F Recurrence relation: a(n)=floor((n-1)/2)*a(n-1)+floor((n+1)/2)*a(n-2); a(1)=0, a(2)=1.
%F G.f.: Q(0)/(x*(1+x)) - 1/x, where Q(k)= 1 + x*(k+1)/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 23 2013
%F Let S=Sum_{n>=0} (1 + x*n + x)*x^(2*n)*(n!)^2, then g.f. A(x)=S/(x+x^2) - 1/x. - _Sergei N. Gladkovskii_, May 23 2013
%F D-finite with recurrence 4*a(n) +2*a(n-1) +(-n^2+n-2)*a(n-2) -n*(n-1)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes and only the vertical one consists only of columns of even length.
%p a[1]:=0: a[2]:=1: for n from 3 to 26 do a[n]:=floor((n-1)/2)*a[n-1]+floor((n+1)/2)*a[n-2] od: seq(a[n],n=1..26);
%Y Cf. A121745, A121749.
%K nonn
%O 1,4
%A _Emeric Deutsch_, Aug 20 2006
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