OFFSET
0,3
COMMENTS
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..448
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 dirigés verticalement convexes, Actes du 31e Séminaire Lotharingien de Combinatoire, Publ. IRMA, Université Strasbourg I (1993).
FORMULA
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^2*a(n-1)-1)/(n-1). - Vladeta Jovovic, Apr 26 2003
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
a(n) = n! + n!*Sum_{j=1..n-1} (n-j)/(j+1)!. - Walden Freedman, Aug 21 2014
Asymptotic approximation: a(n) ~ n!(1 + (n - 1)(e - 2)). - Walden Freedman, Aug 23 2014
EXAMPLE
1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 431*x^5 + 3103*x^6 + 25341*x^7 + 231689*x^8 + ...
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, Aug 07 2006
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1,
(n^2*a(n-1)-1)/(n-1))
end:
seq(a(n), n=0..23); # Alois P. Heinz, Sep 03 2020
MATHEMATICA
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 *)
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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Added a(0) = 1. - Michael Somos, Oct 20 2011
STATUS
approved