A005436 -id:A005436 - cp's OEIS frontend

A002694 Binomial coefficients C(2n, n-2).

1, 6, 28, 120, 495, 2002, 8008, 31824, 125970, 497420, 1961256, 7726160, 30421755, 119759850, 471435600, 1855967520, 7307872110, 28781143380, 113380261800, 446775310800, 1761039350070, 6943526580276, 27385657281648, 108043253365600

Offset: 2

N. J. A. Sloane

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=2. Example: For n=3 there are 6 paths EEENNN, EENENN, EENNEN, EENNNE, ENEENN and NEEENN. - Herbert Kociemba, May 23 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-2 of which are triangular. Example: a(3)=6 because the convex hexagon ABCDEF is dissected by any of the diagonals AC, BD, CE, DF, EA, FB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of UUU's (triple rises), where U=(1,1), in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UD(UUU)DDD, (UUU)DDDUD, (UUU)DUDDD, (UUU)DDUDD and (U[UU)U]DDDD, the triple rises being shown between parentheses. - Emeric Deutsch, Jun 03 2004
Inverse binomial transform of A026389. - Ross La Haye, Mar 05 2005
Sum of the jump-lengths of all full binary trees with n internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given full binary tree is called the jump-length. - Emeric Deutsch, Jan 18 2007
a(n) = number of convex polyominoes (A005436) of perimeter 2n+4 that are directed but not parallelogram polyominoes, because the directed convex polyominoes are counted by the central binomial coefficient binomial(2n,n) and the subset of parallelogram polyominoes is counted by the Catalan number C(n+1) = binomial(2n+2,n+1)/(n+2) and a(n) = binomial(2n,n) - C(n+1). - David Callan, Nov 29 2007
a(n) = number of DUU's in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UU(DUU)DDD, U(DUU)UDDD, U(DUU)DUDD, UDU(DUU)DD, U(DUU)DDUD, UUD(DUU)DD, the DUU's being shown between parentheses and no other Dyck path of semilength 4 contains a DUU. - David Callan, Jul 25 2008
C(2n,n-m) is the number of Dyck-type walks such that their graphs have one marked edge passed 2m times and the other edges are passed 2 times counting "there and back" directions. - Oleksiy Khorunzhiy, Jan 09 2015
Number of paths in the half-plane x >= 0, from (0,0) to (2n,4), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=3, we have the 6 paths: UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, DUUUUU. - José Luis Ramírez Ramírez, Apr 19 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 2..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Henry Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
Milan Janjic, Two Enumerative Functions
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, arXiv:1107.5724 [math-ph], 2014.
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, Zh. Mat. Fiz. Anal. Geom. 10 (1) (2014), pp. 64-125.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012
V. Pilaud and J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv:1307.6440 [math.CO], 2013; Adv. Appl. Math. 57 (2014) 60-100.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
T. Tao and Van Vu, Random matrices: localization of the eigenvalues and the necessity of four moments, arXiv:1005.2901 [math.PR], 2010-2011; Acta Math. Vietnam 36 (2) (2011) 431-449.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Cf. A006659.
Diagonal 5 of triangle A100257.
Cf. A001622, A009766.
Cf. binomial(k*n, n-k): A000027 (k=1), this sequence (k=2), A004321 (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([2..30], n-> Binomial(2*n,n-2)); # G. C. Greubel, Mar 21 2019

a002694 n = a007318' (2 * n) (n - 2)  -- Reinhard Zumkeller, Jun 18 2012

[Binomial(2*n, n-2): n in [2..30]]; // Vincenzo Librandi, Apr 20 2015

a:=n->sum(binomial(n,j-1)*binomial(n,j+1),j=1..n): seq(a(n), n=2..25); # Zerinvary Lajos, Nov 26 2006

CoefficientList[ Series[ 16/(((Sqrt[1 - 4 x] + 1)^4)*Sqrt[1 - 4 x]), {x, 0, 23}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Binomial[2n,n-2],{n,2,30}] (* Harvey P. Dale, Jun 12 2014 *)

{a(n) = binomial(2*n,n-2)}; \\ G. C. Greubel, Mar 21 2019

[binomial(2*n,n-2) for n in (2..30)] # G. C. Greubel, Mar 21 2019

a(n) = A067310(n, 1) as this is number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 1 simple intersection. - Henry Bottomley, Oct 07 2002
E.g.f.: exp(2*x) * BesselI(2, 2*x). - Vladeta Jovovic, Aug 21 2003
G.f.: (1-sqrt(1-4*z))^4/(16*z^2*sqrt(1-4*z)). - Emeric Deutsch, Jan 28 2004
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+2). - Paul Barry, Sep 20 2004
D-finite with recurrence: -(n-2)*(n+2)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012
G.f.: z^2*C(z)^4/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = Sum_{k=1..n} binomial(2*n-k,n-k-1). - Vladimir Kruchinin, Oct 22 2016
G.f.: x^2* 2F1(5/2,3;5;4*x). - R. J. Mathar, Jan 27 2020
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = 23/6 - 13*Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 106*log(phi)/(5*sqrt(5)) - 37/10, where phi is the golden ratio (A001622). (End)
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^2 - 4*x + 2)/sqrt(x*(4 - x)).
G.f. x^2 * B(x) * C(x)^4, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

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.

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, Mar 21 2004

			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)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened).
Mireille Bousquet-Mélou, Convex polyominoes and algebraic languages, Journal of Physics A25 (1992), 1935-1944.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and 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.
K. Y. Lin and S. J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices, Journal of Physics A21 (1988), 2635-2642.

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

[[((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

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

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 *)

{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

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

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

A005770 Number of convex polygons of length 2n on square lattice whose leftmost bottom vertex and rightmost top vertex have the same x-coordinate.

Original entry on oeis.org

1, 9, 55, 286, 1362, 6143, 26729, 113471, 473471, 1951612, 7974660, 32384127, 130926391, 527657073, 2121795391, 8518575466, 34162154550, 136893468863, 548253828965, 2194897467395, 8784784672511, 35153438973304, 140653028240520, 562719731644671

Offset: 5

Simon Plouffe, N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (12,-55,120,-125,54,-8).

Cf. A005436, A005768, A005769.

A005770:=(1-3*z+2*z**2+z**3)/(4*z-1)/(2*z-1)/(z**2-3*z+1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)),{x,0,28}],x] (* Stefano Spezia, Jun 04 2024 *)

a(n) = A005436(n) - A005768(n) - A005769(n).
G.f.: x^5*(1-3*x+2*x^2+x^3)/((1 - 3*x + x^2)^2*(1 - 6*x + 8*x^2)). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003
a(n) = 12*a(n-1) - 55*a(n-2) + 120*a(n-3) - 125*a(n-4) + 54*a(n-5) - 8*a(n-6) for n > 8. - Stefano Spezia, Jun 04 2024

Better description from Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

More terms from Sean A. Irvine, Aug 26 2016

A005768 Number of convex polygons of length 2n on square lattice whose leftmost bottom vertex is strictly to the left of the rightmost top vertex.

Original entry on oeis.org

1, 2, 7, 27, 110, 460, 1948, 8296, 35400, 151056, 643892, 2740216, 11639416, 49340080, 208727176, 881212272, 3713043152, 15615663008, 65555425780, 274734294328, 1149506252376, 4802212126704, 20032666552664, 83452195924304, 347192769717040, 1442672957396320

Offset: 2

Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.

DeleteCases[CoefficientList[Series[x^2*(1 - 8 x + 21 x^2 - 19 x^3 + 4 x^4)/((1 - 2 x^(1/2))^2*(1 + 2 x^(1/2))^2*(1 - 2 x)) - 2 x^4*(1 - 4 x)^(-3/2), {x, 0, 25}], x] , 0] (* Michael De Vlieger, Aug 26 2016 *)

a(n) = A005436(n) - A005769(n) - A005770(n).

G.f.: x^2 * (1 - 8*x + 21*x^2 - 19*x^3 + 4*x^4) / ((1 - 2*x^(1/2))^2 * (1 + 2*x^(1/2))^2*(1 - 2*x)) - 2*x^4*(1 - 4*x)^(-3/2). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

Better description from Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

More terms from Sean A. Irvine, Aug 26 2016

A005769 Number of convex polygons of length 2n on square lattice whose leftmost bottom vertex is strictly to the right of the rightmost top vertex.

Original entry on oeis.org

1, 13, 110, 758, 4617, 25895, 136949, 693369, 3395324, 16197548, 75675657, 347624505, 1574756959, 7051383905, 31266981002, 137492793602, 600295660953, 2604690331787, 11240698270037, 48279130088017, 206486210282936, 879807455701208, 3736101981855305

Offset: 6

Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.

DeleteCases[CoefficientList[Series[x^4*(2 - 20 x + 75 x^2 - 127 x^3 + 95 x^4 - 27 x^5 + 4 x^6)/((1 - 2 x^(1/2))^2*(1 + 2 x^(1/2))^2*(1 - 2 x) (1 + x^(1/2) - x)^2*(1 - x^(1/2) - x)^2) - 2 x^4*(1 - 4 x)^(-3/2), {x, 0, 27}], x] , 0] (* Michael De Vlieger, Aug 26 2016 *)

a(n) = A005436(n) - A005768(n) - A005770(n).

G.f.: x^4 * (2 - 20*x + 75*x^2 - 127*x^3 + 95*x^4 - 27*x^5 + 4*x^6) / ((1 - 2*x^(1/2))^2 * (1 + 2*x^(1/2))^2 * (1 - 2*x) * (1 + x^(1/2) - x)^2 * (1 - x^(1/2) - x)^2) - 2*x^4 * (1 - 4*x)^(-3/2). - Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

Better description from Markus Voege (voege(AT)blagny.inria.fr), Nov 28 2003

More terms from Sean A. Irvine, Aug 26 2016

A128611 Number of Z-convex polyominoes with semiperimeter n.

Original entry on oeis.org

0, 0, 1, 2, 7, 28, 116, 484, 2022, 8448, 35290, 147376, 615228, 2567060, 10704976, 44611804, 185780308, 773060804, 3214225836, 13352979316, 55426067494, 229870371888, 952548347122, 3943943111920, 16316243701350, 67447113649312, 278592165886198, 1149863118820584, 4742473257979906, 19545876370622104, 80502059920697442

Offset: 0

Ralf Stephan, May 08 2007

nonn

Robert Israel, Table of n, a(n) for n = 0..1658
Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See page 2.
E. Duchi, S. Rinaldi and G. Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.

Cf. A003480, A005436.

d:=(1-2*t-sqrt(1-4*t))/2:
t1:=
2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) )
+ t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ):
series(t1,t,120):
seriestolist(%); # N. J. A. Sloane, Oct 02 2011

gf = 2 t^4 (1-2t)^2 d/((1-4t)^2 (1-3t)(1-t)) + t^2 (1-6t+10t^2-2t^3-t^4)/ ((1-4t)(1-3t)(1-t)) /. d -> (1-2t-Sqrt[1-4t])/2;
CoefficientList[gf + O[t]^31, t] (* Jean-François Alcover, Aug 17 2018 *)

The Duchi paper has a g.f.
Asymptotically, a(n) ~ n/24 * 4^n.
G.f.: Let d:=(1-2*t-sqrt(1-4*t))/2; then g.f. is 2*t^4*(1-2*t)^2*d/( (1-4*t)^2*(1-3*t)*(1-t) ) + t^2*(1-6*t+10*t^2-2*t^3-t^4)/( (1-4*t)*(1-3*t)*(1-t) ). - N. J. A. Sloane, Oct 02 2011
(-960+384*n)*a(n)+(1760-992*n)*a(n+1)+(-924+984*n)*a(2+n)+(64-490*n)*a(n+3)+(82+131*n)*a(n+4)+(-24-18*n)*a(n+5)+(2+n)*a(n+6), a(0) = 0. - Robert Israel, Aug 17 2018

A151828 Number of hv-convex sets from class S having semiperimeter n of the bounding rectangle.

Original entry on oeis.org

1, 2, 8, 32, 139, 618, 2779, 12528, 56404, 253152, 1131849, 5040412, 22359981, 98837102

Offset: 2

N. J. A. Sloane, Jul 12 2009

nonn

Peter Balazs, Generation and Empirical Investigation of hv-Convex Discrete Sets, in Image Analysis, Lecture Notes in Computer Science, Volume 4522/2007, Springer-Verlag.

Cf. A005436, A151829.

A151829 Number of hv-convex sets from class S' having semiperimeter n of the bounding rectangle.

Original entry on oeis.org

1, 2, 8, 34, 150, 674, 3056, 13898, 63178, 286570, 1296008, 5842442, 26255254, 117642282

Offset: 2

N. J. A. Sloane, Jul 12 2009

nonn

Peter Balazs, Generation and Empirical Investigation of hv-Convex Discrete Sets, in Image Analysis, Lecture Notes in Computer Science, Volume 4522/2007, Springer-Verlag.

Cf. A005436, A151828.

A260346 Expansion of x^2((1 - 12x + 50x^2 - 76x^3 + 42x^4 - 48x^5 + 32x^6)/(1 - 4x)^4 + 4x^2/(1 - 4x)^(5/2)).

Original entry on oeis.org

0, 0, 1, 4, 22, 124, 706, 3968, 21880, 118192, 625776, 3251744, 16610072, 83537520, 414288080, 2028760544, 9822006896, 47063458528, 223408338400, 1051514839104, 4910856580376, 22772597352944, 104914684398352, 480457417780320, 2188115766353616, 9914318477830304, 44708936142838816

Offset: 0

N. J. A. Sloane, Aug 02 2015

nonn

I. G. Enting and A. J. Guttmann, Area-weighted moments of convex polygons on the square lattice, J. Phys. A 22 (1989), 2639-2642. See Eq. (4).

Cf. A005436.

t1:=x^2*( (1-12*x+50*x^2-76*x^3+42*x^4-48*x^5+32*x^6)/(1-4*x)^4 + 4*x^2/(1-4*x)^(5/2));
series(t1,x,40);
seriestolist(%);

G.f.: x^2*((1 - 12*x + 50*x^2 - 76*x^3 + 42*x^4 - 48*x^5 + 32*x^6)/(1 - 4*x)^4 + 4*x^2/(1 - 4*x)^(5/2)).

A324009 The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Günter Rote, Feb 12 2019

			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 ...

Mireille Bousquet-Mélou, Convex polyominoes and algebraic languages, Journal of Physics A25 (1992), 1935-1944.
M.-P. Delest and G. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Sci., 34 (1984), 169-206.
Ira M. Gessel, On the number of convex polyominoes, Ann. Sci. Math. Québec 24 (2000), no. 1, 63-66.
K. Y. Lin and S. J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices, Journal of Physics A21 (1988), 2635-2642.

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

Row sums are A005436.

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 *)

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) )

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).

cp's OEIS Frontend

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