|
|
A121634
|
|
Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n >= 1; 0 <= k <= n-1).
|
|
2
|
|
|
1, 1, 1, 2, 3, 1, 8, 10, 5, 1, 42, 44, 25, 8, 1, 264, 242, 144, 57, 12, 1, 1920, 1594, 962, 429, 117, 17, 1, 15840, 12204, 7366, 3536, 1131, 219, 23, 1, 146160, 106308, 63766, 32118, 11453, 2664, 380, 30, 1, 1491840, 1036944, 616436, 320710, 123742, 32765, 5704, 620, 38, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
Row sums are the factorials (A000142).
Sum_{k=0..n-1} k*T(n,k) = A121636(n).
The row generating polynomials satisfy P(n,t) = (t+n-2)[(n-2)!+P(n-1,t)] for n >= 3, P(1,t)=1 and P(2,t)=1+t.
|
|
EXAMPLE
|
T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 columns with exactly 2 cells starting at level 0.
Triangle starts:
1;
1, 1;
2, 3, 1;
8, 10, 5, 1;
42, 44, 25, 8, 1;
|
|
MAPLE
|
P[1]:=1: P[2]:=1+t: for n from 3 to 11 do P[n]:=sort(expand((t+n-2)*((n-2)!+P[n-1]))) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
|
|
MATHEMATICA
|
P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);
P[1, _] = 1; P[2, t_] = 1 + t;
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|