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A007808
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Number of directed column-convex polyominoes of height n: a(k+1)=(k+1)*a(k)+(a(1)+...+a(k)).
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10
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1, 1, 3, 13, 69, 431, 3103, 25341, 231689, 2345851, 26065011, 315386633, 4128697741, 58145826519, 876660153671, 14089181041141, 240455356435473, 4343224875615731, 82776756452911579, 1660133837750060001
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirige's verticalement convexes, Actes du 31e Se'minaire Lotharingien de Combinatoire, Publi. IRMA, Universite' Strasbourg I (1993).
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LINKS
| E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos...
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FORMULA
| E.g.f.: (exp(x) - 2 * x) / (1 - x)^2. - Michael Somos, Oct 20 2011
a(n) = A056542(n+1) - A056542(n).
a(n) = (a(n-1)^2 - 2 * a(n-2)^2 + a(n-2) * a(n-3) - 4 * a(n-1) * a(n-3)) / (a(n-2) - a(n-3)) if n>3. - Michael Somos, Oct 20 2011
a(n) = (n^2*a(n-1)-1)/(n-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 26 2003
a(n)=n!*n*[1-Sum(1/(j*(j+1)*(j+1)!), j=1..n-1)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 07 2006
Conjectures from Simon Plouffe, Dec 08 2009: G.f.: (-(x^2+1)*exp(-x)+1)*exp(x)/(-1+x)^2; a(n) = round(n!*n*(exp(1)-2)).
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EXAMPLE
| 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 431*x^5 + 3103*x^6 + 25341*x^7 + 231689*x^8 + ...
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MAPLE
| a:=n->n!*n*(1-add(1/j/(j+1)/(j+1)!, j=1..n-1)): seq(a(n), n=1..22); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 07 2006
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MATHEMATICA
| a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ (Exp[x] - 2 x) / (1 - x)^2, {x, 0, n}] (* Michael Somos, Oct 20 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, n! * polcoeff( (exp(x + x * O(x^n)) - 2 * x) / (1 - x)^2, n))} /* Michael Somos, Oct 20 2011 */
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CROSSREFS
| Cf. A056542.
Sequence in context: A088368 A196794 A184818 * A104989 A119906 A059726
Adjacent sequences: A007805 A007806 A007807 * A007809 A007810 A007811
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KEYWORD
| nonn
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AUTHOR
| Paul.Zimmermann(AT)loria.fr
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EXTENSIONS
| Added a(0) = 1. - Michael Somos, Oct 20 2011
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