A006551 Maximal Eulerian numbers. (Formerly M3426)

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

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]

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.

R. 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

Cf. A008292. Bisections are A025585 and A180056.

Sequence in context: A266386 A134823 A000880 * A353819 A274960 A151826

Adjacent sequences: A006548 A006549 A006550 * A006552 A006553 A006554

Keyword

nonn

Author

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

Status

approved