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%I #4 Jul 26 2022 11:24:07
%S 0,1,5,26,184,1338,11652,108210,1140336,12849714,159858900,2117522754,
%T 30442090248,463511103426,7569181895436,130254363597330,
%U 2383020441932256,45738553437874962,927010880040945924
%N Number of columns of even length in all deco polyominoes of height n. 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)=Sum(k*A121748(n,k), k=0..n-1).
%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)=n*a[n-1]+d(n-1)+(n-1)!*floor((n-1)/2) for n>=2, a(1)=0, where d(1)=1, d(2)=0, d(2n)=3!+5!+...+(2n-1)!, d(2n+1)=-d(2n).
%F Conjecture D-finite with recurrence a(n) +(-n-2)*a(n-1) +(-n^2+4*n-2)*a(n-2) +(n^3-3*n^2-4*n+11)*a(n-3) -(n-1)*(n^2-11*n+26)*a(n-4) +(-n^3+5*n^2+7*n-43)*a(n-5) +(n-3)*(n-5)^2*a(n-6)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 1 and 0 columns of even length, respectively.
%p d:=proc(n) if n=1 then 1 elif n=2 then 0 elif n mod 2 = 0 then add((2*j-1)!,j=2..n/2) else -d(n-1) fi end: a[1]:=0: for n from 2 to 22 do a[n]:=n*a[n-1]+d(n-1)+(n-1)!*floor((n-1)/2) od: seq(a[n],n=1..22);
%Y Cf. A121747, A121748.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Aug 20 2006
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