A121555 Number of 1-cell columns in all deco polyominoes of height n.

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

Table of n, a(n) for n=1..22.

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

Adjacent sequences: A121552 A121553 A121554 * A121556 A121557 A121558

Keyword

nonn,changed

Author

Emeric Deutsch, Aug 08 2006

Status

approved