A275204 Triangle read by rows: T(n,k) = number of domino stacks with n dominoes having a base of k dominoes.

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 8, 7, 1, 1, 7, 15, 12, 9, 1, 1, 9, 22, 25, 16, 11, 1, 1, 10, 31, 43, 35, 20, 13, 1, 1, 12, 41, 68, 65, 45, 24, 15, 1, 1, 13, 54, 99, 113, 87, 55, 28, 17, 1, 1, 15, 66, 143, 178, 159, 109, 65, 32, 19, 1, 1, 16, 82, 193, 273, 267, 205, 131, 75, 36, 21, 1, 1, 18, 98, 258, 398, 430, 357, 251, 153, 85, 40, 23, 1

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

Sequence in context: A135226 A104730 A249488 * A321876 A131238 A133380

Adjacent sequences: A275201 A275202 A275203 * A275205 A275206 A275207

Keyword

nonn,tabl

Author

Tricia Muldoon Brown, Jul 19 2016

Extensions

Data corrected by Jean-François Alcover, Aug 17 2018

Status

approved