login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A093118 Triangle T read by rows: T(m,n) = number of convex polyominoes with an m+1 X n+1 minimal bounding rectangle, m > 0, n <= m. 5
5, 13, 68, 25, 222, 1110, 41, 555, 3951, 19010, 61, 1171, 11263, 70438, 329126, 85, 2198, 27468, 216618, 1245986, 5693968, 113, 3788, 59676, 579330, 4022546, 21832492, 98074332, 145, 6117, 118605, 1389927, 11462495, 72887139, 379145115, 1680306750 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened).

Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.

Ira Gessel, On the number of convex polyominoes, Annales des Sciences Mathématiques du Québec, 24 (2000), 63-66.

V. J. W. Guo and J. Zeng, The number of convex polyominoes and the generating function of Jacobi polynomials, arXiv:math/0403262 [math.CO], 2004.

FORMULA

T(m,n) = ((m+n+m*n)*C(2*m+2*n, 2*m) - 2*m*n*C(m+n, m)^2)/(m+n), for m + n > 0.

T(m,n) = C(2*m+2*n,2*m) + ((2*m+2*n-1)/2)*C(2*m+2*n-2,2*m-1) - 2*(m+n-1) *C(m+n,m)*C(m+n-2,m-1), for m >= 0, n >= 0. - Günter Rote, Feb 12 2019

EXAMPLE

Triangle begins:

   5,

  13,   68,

  25,  222,  1110,

  41,  555,  3951,  19010,

  61, 1171, 11263,  70438,  329126,

  85, 2198, 27468, 216618, 1245986, 5693968,

  ...

This is the lower half of an infinite square table that is symmetric at the main diagonal (T(m,n)=T(n,m)).

From Günter Rote, Feb 12 2019: (Start)

For m=2 and n=1, the T(2,1)=13 polyominoes in a 3 X 2 rectangle are the five polyominoes

.

  +---+---+---+       +---+       +---+---+

  |   |   |   |       |   |       |   |   |

  +---+---+---+   +---+---+---+   +---+---+---+

  |   |   |   |   |   |   |   |       |   |   |

  +---+---+---+   +---+---+---+       +---+---+

.

          +---+           +---+---+

          |   |           |   |   |

          +---+---+---+   +---+---+---+

          |   |   |   |   |   |   |   |

          +---+---+---+   +---+---+---+

.

  plus all their different horizontal and vertical reflections (1 + 2 + 2 + 4 + 4 = 13 polyominoes in total). (End)

MAPLE

T:= (m, n)-> (m+n+m*n)/(m+n)*binomial(2*m+2*n, 2*m)

             -2*m*n/(m+n)*binomial(m+n, m)^2:

seq(lprint(seq(T(m, n), n=1..m)), m=1..10);  # Alois P. Heinz, Feb 24 2019

MATHEMATICA

T[m_, n_] := (m+n+m n)/(m+n) Binomial[2m + 2n, 2m] - 2 m n/(m+n) Binomial[ m+n, m]^2;

Table[T[m, n], {m, 1, 8}, {n, 1, m}] // Flatten (* Jean-François Alcover, Aug 17 2018 *)

PROG

(Sage)

def T(m, n):

         w, h = m+1, n+1 # width and height

         p = w+h         # half the perimeter

         return ( binomial(2*p-4, 2*w-2) + binomial(2*p-6, 2*w-3)*(p-5/2) - 2*(p-3)*binomial(p-2, w-1)*binomial(p-4, w-2) )  # Günter Rote, Feb 13 2019

(PARI) {T(n, k) = ((n+k+n*k)*binomial(2*n+2*k, 2*n) - 2*n*k*binomial(n+k, n)^2)/(n+k)};

for(n=1, 8, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 18 2019

(MAGMA) [[((n+k+n*k)*Binomial(2*n+2*k, 2*n) - 2*n*k*Binomial(n+k, n)^2)/(n+k): k in [1..n]]: n in [1..8]]; // G. C. Greubel, Feb 18 2019

CROSSREFS

Columns T(m, 1) = A001844(m), T(m, 2) = A093119(m). Diagonal T(n, n) = A093120(n).

Sums of T(m,n) with fixed sum m+n (including entries with n > m and the trivial ones: T(0,x)=T(y,0)=1), are A005436. - Günter Rote, Feb 12 2019

Sequence in context: A018678 A149575 A156101 * A087506 A068487 A075063

Adjacent sequences:  A093115 A093116 A093117 * A093119 A093120 A093121

KEYWORD

nonn,tabl,easy,walk

AUTHOR

Ralf Stephan, Mar 21 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 01:33 EST 2019. Contains 329978 sequences. (Running on oeis4.)