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A324009 The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals. 0
1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 25, 68, 25, 1, 1, 41, 222, 222, 41, 1, 1, 61, 555, 1110, 555, 61, 1, 1, 85, 1171, 3951, 3951, 1171, 85, 1, 1, 113, 2198, 11263, 19010, 11263, 2198, 113, 1, 1, 145, 3788, 27468, 70438, 70438, 27468, 3788, 145, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..55.

Ira M. Gessel, On the number of convex polyominoes, Ann. Sci. Math. Québec 24 (2000), no. 1, 63-66.

FORMULA

a(w,h) = binomial(2w+2h-4, 2w-2) + ((2w+2h-5)/2)*binomial(2w+2h-6, 2w-3) - 2(w+h-3)*binomial(w+h-2, w-1)*binomial(w+h-4, w-2), for w > 0, h > 0.

a(w,h) = A093118(w-1,h-1).

EXAMPLE

For w=3 and h=2, the a(3,2)=13 polyominoes spanning a 3 X 2 rectangle are

   XXX   X   XX   X    XX

   XXX  XXX   XX  XXX  XXX

plus all different horizontal and vertical reflections (1+2+2+4+4=13).

Table begins

1  1   1    1   1  1 1 ...

1  5  13   25  41 61 ...

1 13  68  222 555 ...

1 25 222 1110 ...

1 41 555 ...

1 61 ...

1 ...

MATHEMATICA

Table[Binomial[2 # + 2 h - 4, 2 # - 2] + ((2 # + 2 h - 5)/2) Binomial[2 # + 2 h - 6, 2 # - 3] - 2 (# + h - 3) Binomial[# + h - 2, # - 1] Binomial[# + h - 4, # - 2] &[w - h + 1], {w, 10}, {h, w}] // Flatten (* Michael De Vlieger, Apr 15 2019 *)

PROG

(Sage)

def a(w, h):

     s = w+h # half the perimeter

     return ( binomial(2*s-4, 2*w-2) + binomial(2*s-6, 2*w-3)*(s-5/2)

      - 2*(s-3)*binomial(s-2, w-1)*binomial(s-4, w-2) )

CROSSREFS

A093118 contains the same data in a different arrangement and without the entries for w=1 and for h=1.

Row sums are A005436.

Sequence in context: A300035 A130227 A114123 * A143007 A152654 A176487

Adjacent sequences:  A324006 A324007 A324008 * A324010 A324011 A324012

KEYWORD

nonn,easy,tabl,walk

AUTHOR

Günter Rote, Feb 12 2019

STATUS

approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)