A121300 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and having k cells in the longest column (1<=k<=n).

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 12, 5, 1, 1, 31, 35, 15, 6, 1, 1, 63, 95, 48, 18, 7, 1, 1, 127, 249, 145, 58, 21, 8, 1, 1, 255, 640, 418, 181, 68, 24, 9, 1, 1, 511, 1615, 1180, 545, 213, 78, 27, 10, 1, 1, 1023, 4026, 3279, 1593, 649, 245, 88, 30, 11, 1, 1, 2047, 9944, 8981, 4583, 1932, 748, 277, 98, 33, 12, 1

Offset

1,5

Comments

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=1; T(n,2)=2^(n-1)-1=A000225(n-1); T(n,n)=1.

Links

Table of n, a(n) for n=1..78.

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]=Sum(z^i, i=1..k)/[1-Sum(jz^j, j=1..k)] is the g.f. for directed column-convex polyominoes whose columns have height at most k.

Example

Triangle starts:
  1;
  1,1;
  1,3,1;
  1,7,4,1;
  1,15,12,5,1;
  1,31,35,15,6,1;

Maple

f:=k->sum(z^i, i=1..k)/(1-sum(j*z^j, j=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 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Mathematica

f[k_] := Sum[z^i, {i, 1, k}]/(1 - Sum[j*z^j, {j, 1, k}]);
T[n_, k_] := If[k <= n, SeriesCoefficient[f[k] - f[k-1], {z, 0, n}], 0];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 25 2024, after Maple program *)

Crossrefs

Cf. A001519, A000225.

Sequence in context: A126713 A140068 A179745 * A283595 A128119 A158198

Adjacent sequences: A121297 A121298 A121299 * A121301 A121302 A121303

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 04 2006

Status

approved