A006027 Number of directed column-convex polyominoes with perimeter 2n+2. (Formerly M1647)

1, 1, 2, 6, 20, 71, 263, 1005, 3933, 15684, 63505, 260390, 1079019, 4511700, 19011521, 80653480, 344193353, 1476589475, 6364258163, 27545933212, 119676949397, 521739175908, 2281673067934, 10006784399183, 44002280467770, 193957104163645, 856853526774173

Offset

1,3

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

M.-P. Delest, Utilisation des Langages Algébriques et du Calcul Formel Pour le Codage et l'Enumeration des Polyominos, Ph.D. Dissertation, Université Bordeaux I, May 1987. [Scanned copy, with permission. A very large file.]

M.-P. Delest, Utilisation des Langages Algébriques et du Calcul Formel Pour le Codage et l'Enumeration des Polyominos, Ph.D. Dissertation, Université Bordeaux I, May 1987. (Annotated scanned copy of a small part of the thesis)

M.-P. Delest, Generating functions for column-convex polyominoes, J. Combin. Theory Ser. A 48 (1988), no. 1, 12-31.

M.-P. Delest and S. Dulucq, Enumeration of directed column-convex animals with given perimeter and area, Croat. Chem. Acta. 66 (1993), 59-80.

E. Duchi and S. Rinaldi, An object grammar for column-convex polyominoes, Annals of Combinatorics, 8 (2004), 27-36.

S. Dulucq, Etude combinatoire de problèmes d'énumeration, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)

Formula

G.f. A(x) = a(1)x^2 + a(2)x^3 + a(3)x^4 + ... satisfies the functional equation A^3 + 2(x-1)A^2 + (2x-1)(x-1)A + (x^2)(x-1) = 0. - D. G. Rogers, May 22 2005

Mathematica

m = 30; A[_] = 0;
Do[A[x_] = (2(1-x) A[x]^2 - A[x]^3 + x^2 - x^3)/((1-x)(1-2x))+O[x]^m, {m}];
CoefficientList[1 + A[x]/x, x] (* Jean-François Alcover, Oct 05 2019 *)

Crossrefs

Cf. A005435.

Sequence in context: A108600 A274484 A128729 * A049124 A275756 A301627

Adjacent sequences: A006024 A006025 A006026 * A006028 A006029 A006030

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from D. G. Rogers and Emanuele Munarini, May 15 2005

Status

approved