A121301 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and having k cells in the shortest column (1<=k<=n).

1, 1, 1, 4, 0, 1, 10, 2, 0, 1, 28, 5, 0, 0, 1, 75, 10, 3, 0, 0, 1, 202, 23, 7, 0, 0, 0, 1, 540, 57, 8, 4, 0, 0, 0, 1, 1440, 129, 18, 9, 0, 0, 0, 0, 1, 3828, 294, 43, 10, 5, 0, 0, 0, 0, 1, 10153, 680, 90, 11, 11, 0, 0, 0, 0, 0, 1, 26875, 1557, 178, 28, 12, 6, 0, 0, 0, 0, 0, 1, 71021, 3546, 362, 69, 13, 13, 0, 0, 0, 0, 0, 0, 1

Offset

1,4

Comments

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=fibonacci(2n-1)-A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n and having k 1-cell columns). T(n,n)=1.

Links

Table of n, a(n) for n=1..91.

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

Formula

G.f. of column k is f[k]-f[k+1], where f[k]=z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(1+k)) is the g.f. for directed column-convex polyominoes whose columns have height at least k.

Example

Triangle starts:
  1;
  1,1;
  4,0,1;
  10,2,0,1;
  28,5,0,0,1;
  75,10,3,0,0,1;
  202,23,7,0,0,0,1;

Maple

f:=k->z^k*(1-z)/(z^2-2*z+1+z^(1+k)*k-k*z^k-z^(1+k)): T:=proc(n, k) if k<=n then coeff(series(f(k)-f(k+1), z=0, 15), z, n) else 0 fi end: for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Mathematica

f[k_] := z^k*(1 - z)/(z^2 - 2*z + 1 + z^(1 + k)*k - k*z^k - z^(1 + k));
T[n_, k_] := If[k <= n, SeriesCoefficient[f[k] - f[k+1], {z, 0, n}], 0];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 25 2024, after Maple Program *)

Crossrefs

Cf. A001519, A121469, A121300.

Sequence in context: A189245 A342372 A289222 * A059056 A344393 A127153

Adjacent sequences: A121298 A121299 A121300 * A121302 A121303 A121304

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 04 2006

Status

approved