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A121747
Number of columns of odd 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
1, 2, 8, 44, 262, 1938, 15600, 145086, 1461888, 16438446, 198598860, 2623055166, 36933441912, 560697617214, 9014444990964, 154698782105070, 2795947673216544, 53529558912435438, 1074325981318055676, 22703327631819203166, 500976898248191064840
OFFSET
1,2
LINKS
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} k * A121745(n,k).
Recurrence relation: a(n)=n*a[n-1]-d(n-1)+(n-1)!*floor(n/2) for n>=2, a(1)=1, where d(1)=1, d(2)=0, d(2n)=3!+5!+...+(2n-1)!, d(2n+1)=-d(2n).
Conjecture D-finite with recurrence (-9*n+38)*a(n) +3*(3*n^2-19*n+27)*a(n-1) +(9*n^3-37*n^2-24*n+127)*a(n-2) +(-9*n^4+102*n^3-392*n^2+525*n-85)*a(n-3) -(n-3)*(28*n^3-223*n^2+510*n-237)*a(n-4) -(19*n-15)*(n-3)*(n-4)^2*a(n-5)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns of odd 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]:=1: for n from 2 to 22 do a[n]:=n*a[n-1]-d(n-1)+(n-1)!*floor(n/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, 1, n*a[n-1] - d[n-1] + (n-1)!*Floor[n/2]];
Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Aug 20 2024, after Maple program *)
CROSSREFS
Sequence in context: A291697 A047851 A177260 * A261266 A014508 A141147
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 20 2006
STATUS
approved