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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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11
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1, 1, 3, 13, 69, 431, 3103, 25341, 231689, 2345851, 26065011, 315386633, 4128697741, 58145826519, 876660153671, 14089181041141, 240455356435473, 4343224875615731, 82776756452911579, 1660133837750060001, 34950186057896000021, 770651602576606800463
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OFFSET
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0,3
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COMMENTS
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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
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 3, 4, etc., along the main diagonal, and zeros everywhere else. Then equals a(n) equals the permanent of M(n-1) for n >= 2. - John M. Campbell, Apr 20 2021
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LINKS
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FORMULA
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E.g.f.: (exp(x) - 2 * x) / (1 - x)^2. - Michael Somos, Oct 20 2011
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!*n*(1-Sum_{j=1..n-1} 1/(j*(j+1)*(j+1)!)). - Emeric Deutsch, Aug 07 2006
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
Asymptotic approximation: a(n) ~ n!(1 + (n - 1)(e - 2)). - Walden Freedman, Aug 23 2014
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EXAMPLE
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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
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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
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1,
(n^2*a(n-1)-1)/(n-1))
end:
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MATHEMATICA
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a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ (Exp[x] - 2 x) / (1 - x)^2, {x, 0, n}]] (* Michael Somos, Oct 20 2011 *)
a[n_] := n! + n!*Sum[(n - j)/(j + 1)!, {j, 1, n - 1}] (* Walden Freedman, Aug 21 2014 *)
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PROG
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(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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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