A122105 Sum of the bottom levels of all columns 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.

0, 0, 0, 1, 11, 101, 932, 9080, 94852, 1066644, 12905784, 167622984, 2330016768, 34551794304, 544873631616, 9110134903680, 161038110977280, 3001678242428160, 58853489050759680, 1211082030609016320, 26101332373130496000, 588033071962511616000

Offset

0,5

Links

Table of n, a(n) for n=0..21.

E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Formula

a(n) = Sum_{k>=0} k*A122104(n,k).

Recurrence relation: a(n) = (2n-1)*a(n-1)-(n-1)^2*a(n-2)+(n-2)!*(n-2) for n>=3, a(0)=a(1)=a(2)=0.

a(n) = n![n - H(n) - (H(n))^2/2 + (1/2)Sum(1/j^2, j=1..n)], where H(n)=Sum(1/j, j=1..n). - Emeric Deutsch, Apr 06 2008

E.g.f.: (2 * x + (1 - x) * log(1 - x) * (2 - log(1 - x))) / (2 * (1 - x)^2). - Ilya Gutkovskiy, Sep 02 2021

D-finite with recurrence a(n) +(-3*n+1)*a(n-1) +(3*n^2-4*n-2)*a(n-2) +(-n^3+2*n^2+7*n-15)*a(n-3) +(n-3)^3*a(n-4)=0. - R. J. Mathar, Jul 26 2022

Example

a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having all their columns starting at level zero.

Maple

a[0]:=0: a[1]:=0: a[2]:=0: for n from 3 to 23 do a[n]:=(2*n-1)*a[n-1]-(n-1)^2*a[n-2]+(n-2)*(n-2)! od: seq(a[n], n=0..23);

Mathematica

RecurrenceTable[{a[0]==a[1]==0, a[n]==(2n-1)*a[n-1]-(n-1)^2*a[n-2]+(n-2)!*(n-2)}, a, {n, 0, 20}] (* Harvey P. Dale, Dec 04 2014; adapted to offset 0 by Georg Fischer, Jul 30 2022 *)

Crossrefs

Cf. A122104.

Sequence in context: A037550 A138894 A287832 * A332853 A289850 A351811

Adjacent sequences: A122102 A122103 A122104 * A122106 A122107 A122108

Keyword

nonn

Author

Emeric Deutsch, Aug 24 2006

Status

approved