|
| |
|
|
A006551
|
|
Maximal Eulerian numbers.
(Formerly M3426)
|
|
5
| |
|
|
1, 1, 4, 11, 66, 302, 2416, 15619, 156190, 1310354, 15724248, 162512286, 2275172004, 27971176092, 447538817472, 6382798925475, 114890380658550, 1865385657780650, 37307713155613000, 679562217794156938, 14950368791471452636
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| Contribution 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)
|
|
|
REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| Digital Library of Mathematical Functions, Table 26.14.1 [From Peter Luschny, Aug 08 2010]
|
|
|
FORMULA
| a(n) = sum_{0<=j<=floor(n/2)} (-1)^j binomial(n+1,j) (floor(n/2)-j+1)^n [From Peter Luschny, Aug 08 2010]
|
|
|
MAPLE
| Contribution from Peter Luschny, Aug 08 2010: (Start)
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: (End)
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
|
|
|
CROSSREFS
| Cf. A008292. Bisections are A025585 and A180056.
Sequence in context: A114053 A134823 A000880 * A151826 A032110 A054234
Adjacent sequences: A006548 A006549 A006550 * A006552 A006553 A006554
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)
|
| |
|
|
