A006027 Number of directed column-convex polyominoes with perimeter 2n+2.

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

Author

Simon Plouffe

Keywords

• nonn

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)

Crossrefs

Cf. A005435.

Programs

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 *)

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

Extensions

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