OFFSET
1,3
COMMENTS
a(n) = A180190(n,1).
a(n+2) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = k! for k = 1, ..., n+1. - Michael Somos, Jan 05 2012
From Jon Perry, Jan 04 2013: (Start)
Number of permutations of {1,...,n-1,n+1} with at least one indexed point p(k)=k with 1<=k<=n. Note that this means p(k)=n+1 is never an indexed point as k<n+1. Permutations of {1,2,4} with an indexed point p(k)=k are 124, 142 and 421, so a(3)=3.
For n>1, a(n) is the number of permutations of [n+1] that have a fixed point and contain 12; for example the a(3)=3 such permutations of {1,2,3,4} are 1234, 1243, and 3124.
(End)
For n > 0: row sums of triangle A116853. - Reinhard Zumkeller, Aug 31 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..200
Uriel Feige, Tighter bounds for online bipartite matching, 2018.
FORMULA
EXAMPLE
x^2 + 3*x^3 + 13*x^4 + 67*x^5 + 411*x^6 + 2921*x^7 + 23633*x^8 + ...
a(3) = 3 because we have 123, 312, and 231; the permutations 132, 213, and 321 have no successions.
a(4) = 13 since p(x) = (3*x^2 - 7*x + 6) / 2 interpolates p(1) = 1, p(2) = 2, p(3) = 6, and p(4) = 13. - Michael Somos, Jan 05 2012
MAPLE
d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: seq(factorial(n)-d[n]-d[n-1], n = 1 .. 22);
MATHEMATICA
f[n_] := Sum[ -(-1)^k (n - k)! Binomial[n - 1, k], {k, 1, n}]; Array[f, 20] (* Robert G. Wilson v, Oct 16 2010 *)
PROG
(PARI) {a(n) = if( n<2, 0, n--; subst( polinterpolate( vector( n, k, k!)), x, n+1))} /* Michael Somos, Jan 05 2012 */
(Haskell)
a180191 n = if n == 1 then 0 else sum $ a116853_row (n - 1)
-- Reinhard Zumkeller, Aug 31 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 07 2010
STATUS
approved