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A121582
Number of cells in column 2 of 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.
3
0, 1, 7, 40, 252, 1837, 15259, 141798, 1455694, 16360387, 199845957, 2637020884, 37388864368, 566971338009, 9157693715407, 156975522127762, 2846305448882274, 54432896145210943, 1095019542858729769
OFFSET
1,3
COMMENTS
a(n)=Sum(k*A121581(n,k),k=0..n-1).
REFERENCES
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
LINKS
FORMULA
a(1)=0, a(2)=1, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)+(n-1)!(n-2)(n^2-3n+4)/2]/(n-2) for n>=3.
a(n) ~ n*n!/2. - Vaclav Kotesovec, Aug 15 2013
D-finite with recurrence (-49*n+454)*a(n) +(49*n^2-454*n-1328)*a(n-1) +(49*n^2+1553*n-1464)*a(n-2) +(-581*n^2+612*n+1035)*a(n-3) +(819*n^2-3682*n+4007)*a(n-4) -4*(84*n-169)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
MAPLE
a:=proc(n) if n=1 then 0 elif n=2 then 1 else ((2*n-3)*a(n-1)-(n-1)*a(n-2)+(n-1)!*(n-2)*(n^2-3*n+4)/2)/(n-2) fi end: seq(a(n), n=1..22);
MATHEMATICA
RecurrenceTable[{a[1]==0, a[2]==1, a[n]==((2n-3)a[n-1]-(n-1)a[n-2]+ (n-1)!(n-2) (n^2-3n+4)/2)/(n-2)}, a, {n, 20}] (* Harvey P. Dale, Oct 23 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 11 2006
STATUS
approved