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A262033
Number of permutations of [n] beginning with at least floor(n/2) ascents.
5
1, 1, 1, 3, 4, 20, 30, 210, 336, 3024, 5040, 55440, 95040, 1235520, 2162160, 32432400, 57657600, 980179200, 1764322560, 33522128640, 60949324800, 1279935820800, 2346549004800, 53970627110400, 99638080819200, 2490952020480000, 4626053752320000
OFFSET
0,4
LINKS
FORMULA
E.g.f.: (x+1)*(exp(x^2)-1)/x^2.
a(n) = 2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)) for n>1, a(0)=a(1)=1.
a(n) = n!/ceiling((n+1)/2)!.
a(2n) = A262034(2n) = A001761(n).
a(2n+1) = A006963(n+2).
Sum_{n>=0} 1/a(n) = 7/4 + 13*exp(1/4)*sqrt(Pi)*erf(1/2)/8, where erf is the error function. - Amiram Eldar, Dec 04 2022
EXAMPLE
a(4) = 4: 1234, 1243, 1342, 2341.
a(5) = 20: 12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 23415, 23451, 23514, 23541, 24513, 24531, 34512, 34521.
MAPLE
a:= proc(n) option remember; `if`(n<2, 1,
2*n*(n*(n-1)*a(n-2)-a(n-1))/((n+2)*(n-1)))
end:
seq(a(n), n=0..30);
MATHEMATICA
a[n_] := n!/Ceiling[(n + 1)/2]!; Array[a, 30, 0] (* Amiram Eldar, Dec 04 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 08 2015
STATUS
approved