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

Author

Emeric Deutsch, Aug 08 2006

Keywords

• nonn

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]

Examples

			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.
		

Links

• E. Barcucci, A. Del Lungo, and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.

Crossrefs

Cf. A121554.

Programs

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 *)

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