

A275204


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


3



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
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OFFSET

1,5


COMMENTS

The kth column is the number of domino stacks having a base of k dominoes.
The nth 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*(ki)+1)*T(nk,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/(1x^k) Sum_{S} (Product_{i=1..j} (2*(k_{i+1}k_i)+1)*x^(k_i)/ (1x^(k_i))) where the sum is over all subsets S of {1,..,k1} such that S={k_1<k_2<..<k_{j1}} 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*(ki)+1)*T(nk, 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 (ki) + 1) T[nk, i], {i, 1, k}];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* JeanFrançois 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 JeanFrançois Alcover, Aug 17 2018


STATUS

approved



