A006551 Maximal Eulerian numbers.

1, 1, 4, 11, 66, 302, 2416, 15619, 156190, 1310354, 15724248, 162512286, 2275172004, 27971176092, 447538817472, 6382798925475, 114890380658550, 1865385657780650, 37307713155613000, 679562217794156938, 14950368791471452636

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn

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

Crossrefs

Cf. A008292. Bisections are A025585 and A180056.

Programs

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 *)

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