OFFSET
1,5
COMMENTS
The k-th column is the number of domino stacks having a base of k dominoes.
The n-th row is the number of domino stacks consisting of n dominoes.
A domino stack corresponds to a convex polyomino built with dominoes such that all columns intersect the base.
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
T. M. Brown Convex Domino Towers, J. Integer Seq. 20 (2017).
FORMULA
T(n,k) = Sum_{i=1..k} (2*(k-i)+1)*T(n-k,i) where T(n,n)=1 and T(n,k)=0 if n or k is nonpositive or if n is less than k.
G.f.: x^k/(1-x^k) Sum_{S} (Product_{i=1..j} (2*(k_{i+1}-k_i)+1)*x^(k_i)/ (1-x^(k_i))) where the sum is over all subsets S of {1,..,k-1} such that S={k_1<k_2<..<k_{j-1}} and k_j=k.
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 4, 5, 1;
1, 6, 8, 7, 1;
1, 7, 15, 12, 9, 1;
...
Dominoes are assumed to be horizontal, and each row must be a subset of the row below it. For n=4 and k=2, the bottom row has 2 dominoes. One possibility is to put both remaining dominoes in the next row up. Otherwise there will be one domino in the next row up, and it can be in three possible positions: right, center, or left. The last domino must be placed on top of it. So there are a total of four possible stacks, and therefore T(4,2) = 4. - Michael B. Porter, Jul 20 2016
MAPLE
T:= proc(n, k) option remember; `if`(k<1 or k>n, 0,
`if`(n=k, 1, add((2*(k-i)+1)*T(n-k, i), i=1..k)))
end:
seq(seq(T(n, k), k=1..n), n=1..15); # Alois P. Heinz, Jul 19 2016
MATHEMATICA
T[n_, n_] = 1; T[n_, k_] /; n<1 || k<1 || n<k = 0;
T[n_, k_] := T[n, k] = Sum[(2 (k-i) + 1) T[n-k, i], {i, 1, k}];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 17 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tricia Muldoon Brown, Jul 19 2016
EXTENSIONS
Data corrected by Jean-François Alcover, Aug 17 2018
STATUS
approved