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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 5, 3, 1, 87, 20, 8, 4, 1, 567, 107, 28, 12, 5, 1, 4167, 674, 135, 40, 17, 6, 1, 34407, 4841, 809, 175, 57, 23, 7, 1, 316647, 39248, 5650, 984, 232, 80, 30, 8, 1, 3219687, 355895, 44898, 6634, 1216, 312, 110, 38, 9, 1, 35878887, 3575582, 400793, 51532, 7850, 1528, 422, 148, 47, 10, 1
Offset: 1

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Author

Emeric Deutsch, Jun 03 2008

Keywords

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.

Examples

			T(2,1)=1 (the vertical domino); T(2,2)=1 (the horizontal domino); T(3,1)=3 because we have (3), (1,2) and (2,1,1), where (a,b,c,...) stands for a polyomino with columns of lengths a,b,c,..., starting at level 0.
Triangle starts:
    1;
    1,   1;
    3,   2,   1;
   15,   5,   3,   1;
   87,  20,   8,   4,   1;
  567, 107,  28,  12,   5,   1;
		

Crossrefs

Programs

  • Maple
    T:=proc(n,k) options operator, arrow: binomial(n-1, k-1)+sum(factorial(j)*(j-1)*binomial(n-1-j, k-1),j=2..n-1) end proc: for n to 11 do seq(T(n, k),k=1..n) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==1, n! -Sum[j!, {j,n-1}], T[n-1, k] + T[n-1, k-1] ]];
    Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, May 02 2021 *)
  • PARI
    T(n,k) = binomial(n-1, k-1) + sum(j=2, n-1, j!*(j-1)*binomial(n-1-j, k-1));
    tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 16 2019
    
  • Sage
    @CachedFunction
    def T(n, k):
        if (k < 0 or k > n): return 0
        elif (k==1): return factorial(n) - sum(factorial(j) for j in (1..n-1))
        else: return T(n-1, k-1) + T(n-1, k)
    flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 02 2021

Formula

T(n,k) = binomial(n-1, k-1) + Sum_{j=2..n-1} j!*(j-1)*binomial(n-1-j, k-1).
T(n,k) = T(n-1, k) + T(n-1, k-1) for n,k >= 2.
Sum of entries in row n is n! (A000142).
T(n,1) = A132371(n).
Sum_{k=1..n} k*T(n,k) = A140710(n).