A121633 Sum of the bottom levels of the last column over 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, 1, 9, 68, 527, 4408, 40303, 403046, 4393339, 51955528, 663383135, 9102982354, 133668773755, 2092209897524, 34783032728383, 612234346270510, 11375905660965179, 222544581264066400, 4572536725690159999, 98456173247669999978, 2217126753620449439515

Offset: 1

Author

Emeric Deutsch, Aug 12 2006

Keywords

• nonn

Comments

a(n) = Sum(k*A121632(n,k), k>=0).

Examples

			a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, all of whose columns start at level 0.
		

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..449

Crossrefs

Cf. A121632, A000254.

Programs

Maple

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

Mathematica

RecurrenceTable[{a[1]==0,a[n]==n*a[n-1]+(n-1)!-1},a,{n,20}] (* Harvey P. Dale, Dec 01 2013 *)

Formula

a(1)=0; a(n) = n*a(n-1)+(n-1)!-1 for n>=2.

a(n)= A000254(n)- A002672(n) a(n)= n!*sum(1/k,k=1..10)- floor(n!(e-1)) [From Gary Detlefs, Jul 18 2010]

D-finite with recurrence a(n) +(-2*n-1)*a(n-1) +(n^2+2*n-4)*a(n-2) +(-2*n^2+6*n-3)*a(n-3) +(n-3)^2*a(n-4)=0. - R. J. Mathar, Jul 26 2022