OFFSET
1,3
COMMENTS
From Peter Luschny, Aug 08 2010: (Start)
Define A(n,k) as the number of permutations of {1,2,..,n} with k ascents.
A(n,k) = sum_{j=0}^k (-1)^j binomial(n+1,j)(k-j+1)^n.
Then a(n) = A(n, floor(n/2)). The Digital Library of Mathematical Functions calls the A(n,k) Eulerian numbers. With this terminology a(n) are the middle Eulerian numbers and A180056 the central Eulerian numbers. (End)
Number of permutations of {1,2,..,n} with floor(n/2) descents. - Joerg Arndt, Aug 15 2014
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..450
Digital Library of Mathematical Functions, Table 26.14.1 [Peter Luschny, Aug 08 2010]
Herman Chau, On Enumerating Higher Bruhat Orders Through Deletion and Contraction, arXiv:2412.10532 [math.CO], 2024. See p. 20.
L. Lesieur and J.-N. Nicolas, On the Eulerian numbers M_n = max_{1<=k<=n} A(n,k), European J. Combin., 13 (1992), 379-399.
Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994
FORMULA
a(n) = sum_{0<=j<=floor(n/2)} (-1)^j binomial(n+1,j) (floor(n/2)-j+1)^n. [Peter Luschny, Aug 08 2010]
a(n+1)/a(n) ~ n. - Ran Pan, Oct 26 2015
a(n) ~ 2 * sqrt(3) * n^n / exp(n). - Vaclav Kotesovec, Oct 28 2021
MAPLE
a := proc(n) local j, k; k := iquo(n, 2);
add((-1)^j*binomial(n+1, j)*(k-j+1)^n, j=0..k) end:
# Peter Luschny, Aug 08 2010
# Computation by recursion:
A006551 := proc(r) local W; W := proc(m) local A, n, k;
A:=[seq(1, n=1..m)]; if m < 2 then RETURN(1) fi;
for n from 2 to m-1 do for k from 2 to m do
A[k] := n*A[k-1]+k*A[k] od od; [A[m-1], A[m]] end:
W((r+2+irem(r, 2))/2)[2-irem(r, 2)] end:
# Peter Luschny, Jan 12 2011
MATHEMATICA
a[n_] := With[{k = Quotient[n, 2]}, Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k}]]; Array[a, 25] (* Jean-François Alcover, Feb 19 2017, after Peter Luschny *)
CROSSREFS
KEYWORD
nonn,changed
STATUS
approved