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 A121750 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. 2
 0, 1, 5, 26, 184, 1338, 11652, 108210, 1140336, 12849714, 159858900, 2117522754, 30442090248, 463511103426, 7569181895436, 130254363597330, 2383020441932256, 45738553437874962, 927010880040945924 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Table of n, a(n) for n=1..19. Elena Barcucci, Sara Brunetti, and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14. Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42. FORMULA a(n) = Sum_{k=0..n-1} k*A121748(n,k). 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). 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 EXAMPLE 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. MAPLE 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); MATHEMATICA d[n_] := Which[n == 1, 1, n == 2, 0, EvenQ[n], Sum[(2j - 1)!, {j, 2, n/2}], True, -d[n-1]]; a[n_] := a[n] = If[n == 1, 0, n*a[n-1] + d[n-1] + (n-1)!*Floor[(n-1)/2]]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Aug 20 2024, after Maple program *) CROSSREFS Cf. A121747, A121748. Sequence in context: A302896 A368176 A175151 * A143341 A303970 A366215 Adjacent sequences: A121747 A121748 A121749 * A121751 A121752 A121753 KEYWORD nonn AUTHOR Emeric Deutsch, Aug 20 2006 STATUS approved

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Last modified September 7 22:13 EDT 2024. Contains 375749 sequences. (Running on oeis4.)