login
A121555
Number of 1-cell columns in all deco polyominoes of height n.
2
1, 2, 7, 32, 178, 1164, 8748, 74304, 704016, 7362720, 84255840, 1047358080, 14054739840, 202514376960, 3118666924800, 51119166873600, 888640952371200, 16330301780889600, 316322420114534400, 6441691128993792000, 137586770616637440000, 3075566993729556480000
OFFSET
1,2
COMMENTS
A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
It appears that a(n) is a function of the harmonic numbers. [Gary Detlefs, Aug 13 2010]
LINKS
E. Barcucci, A. Del Lungo, and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8.
FORMULA
a(n) = Sum_{k=0..n} k*A121554(n, k).
a(1) = 1, a(n) = n*a(n-1)+(n-2)!*(n-2) for n >= 2.
a(n) = n!*(h(n) - (n-1)/n), where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Aug 13 2010
(-n+3)*a(n) + (2*n^2-7*n+4)*a(n-1) - (n-1)*(n-2)^2*a(n-2) = 0. - R. J. Mathar, Jul 15 2017
a(n) = abs(Stirling1(n + 1, 2)) - (n - 1)*(n - 1)!. - Detlef Meya, Apr 09 2024
EXAMPLE
a(2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
MAPLE
a[1]:=1: for n from 2 to 23 do a[n]:=n*a[n-1]+(n-2)!*(n-2) od:
seq(a[n], n = 1..23);
# Alternative:
a := n -> (n - 1)! * (n*harmonic(n) - (n - 1)):
seq(a(n), n = 1..22); # Peter Luschny, Apr 09 2024
MATHEMATICA
a[n_]:=Abs[StirlingS1[n+1, 2]]-(n-1)*(n-1)!; Flatten[Table[a[n], {n, 1, 22}]] (* Detlef Meya, Apr 09 2024 *)
CROSSREFS
Cf. A121554.
Sequence in context: A059439 A190123 A006014 * A265165 A351813 A301465
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 08 2006
STATUS
approved