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A007808 Number of directed column-convex polyominoes of height n: a(k+1)=(k+1)*a(k)+(a(1)+...+a(k)). 11

%I

%S 1,1,3,13,69,431,3103,25341,231689,2345851,26065011,315386633,

%T 4128697741,58145826519,876660153671,14089181041141,240455356435473,

%U 4343224875615731,82776756452911579,1660133837750060001,34950186057896000021,770651602576606800463

%N Number of directed column-convex polyominoes of height n: a(k+1)=(k+1)*a(k)+(a(1)+...+a(k)).

%C a(n) is also the number of outcomes to a race with n contestants in which there is at most one tie (of at least two contestants). - _Walden Freedman_, Aug 21 2014

%H Alois P. Heinz, <a href="/A007808/b007808.txt">Table of n, a(n) for n = 0..448</a>

%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.

%H E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, <a href="http://www.emis.de/journals/SLC/opapers/s31barc.html">La hauteur des polyominos dirigés verticalement convexes</a>, Actes du 31e Séminaire Lotharingien de Combinatoire, Publ. IRMA, Université Strasbourg I (1993).

%F E.g.f.: (exp(x) - 2 * x) / (1 - x)^2. - _Michael Somos_, Oct 20 2011

%F a(n) = A056542(n+1) - A056542(n).

%F 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

%F a(n) = (n^2*a(n-1)-1)/(n-1). - _Vladeta Jovovic_, Apr 26 2003

%F a(n) = n!*n*[1-Sum(1/(j*(j+1)*(j+1)!), j=1..n-1)). - _Emeric Deutsch_, Aug 07 2006

%F Conjectures: E.g.f.: (-(x^2+1)*exp(-x)+1)*exp(x)/(-1+x)^2; a(n) = round(n!*n*(exp(1)-2)). - _Simon Plouffe_, Dec 08 2009

%F a(n) = n! + n!*sum(j=1..n-1, (n-j)/(j+1)! ). - _Walden Freedman_, Aug 21 2014

%F Asymptotic approximation: a(n) ~ n!(1 + (n - 1)(e - 2)). - _Walden Freedman_, Aug 23 2014

%e 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 431*x^5 + 3103*x^6 + 25341*x^7 + 231689*x^8 + ...

%p a:=n->n!*n*(1-add(1/j/(j+1)/(j+1)!,j=1..n-1)): seq(a(n),n=1..22); # _Emeric Deutsch_, Aug 07 2006

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<2, 1,

%p (n^2*a(n-1)-1)/(n-1))

%p end:

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Sep 03 2020

%t a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ (Exp[x] - 2 x) / (1 - x)^2, {x, 0, n}]] (* _Michael Somos_, Oct 20 2011 *)

%t a[n_] := n! + n!*Sum[(n - j)/(j + 1)!, {j, 1, n - 1}] (* _Walden Freedman_, Aug 21 2014 *)

%o (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 */

%Y Cf. A056542.

%K nonn

%O 0,3

%A _Paul Zimmermann_

%E Added a(0) = 1. - _Michael Somos_, Oct 20 2011

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Last modified April 12 21:47 EDT 2021. Contains 342933 sequences. (Running on oeis4.)