login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A025585 Central Eulerian numbers A(2n-1,n). 5
1, 4, 66, 2416, 156190, 15724248, 2275172004, 447538817472, 114890380658550, 37307713155613000, 14950368791471452636, 7246997577257618116704, 4179647109945703200884716, 2828559673553002161809327536, 2219711218428375098854998661320 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

B. Sturmfels, Solving Systems of Polynomial Equations, Amer. Math. Soc., 2002, see p. 27 (is that the same sequence?)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..200

David H. Bailey and Jonathan M. Borwein, Experimental computation with oscillatory integrals, Comtemp. Math. 517 (2010), 25-40, MR 2731059. [Added by N. J. A. Sloane, Nov 02 2009]

FORMULA

a(n) = sum((-1)^j*(n-j)^(2n-1)*binomial(2n, j), j=0..n). This is T(2n-1, n), where T(n, k) = sum((-1)^j*(k-j+1)^n*binomial(n+1, j), j=0..k) (Cf. A008292. and http://dlmf.nist.gov/26.14#T1)

a(n) = 2*n* A180056(n-1). - Gary Detlefs, Nov 11 2011

a(n+1)/a(n) ~ 4*n^2. - Ran Pan, Oct 26 2015

a(n) ~ sqrt(3) * 2^(2*n) * n^(2*n-1) / exp(2*n). - Vaclav Kotesovec, Oct 16 2016

MAPLE

# First program

A025585 := n-> add((-1)^j *(n-j)^(2*n-1) *binomial (2*n, j), j=0..n-1):

seq(A025585(n), n=1..30);

# This second program computes the list of

# the first m Central Eulerian numbers very efficiently

A025585_list :=

   proc(m) local A, R, n, k;

      R := 1;

      if m > 1 then

         A := array([seq(1, n=1..m)]);

         for n from 2 to m do

            for k from 2 to m do

               A[k] := n*A[k-1] + k*A[k];

               if n = k then R:= R, A[k] fi

            od

         od

      fi;

      R

   end:

A025585_list(30); # Peter Luschny, Jan 11 2011

MATHEMATICA

f[n_] := Sum[(-1)^j*(n - j)^(2 n - 1)*Binomial[2 n, j], {j, 0, n}]; Array[f, 14] (* Robert G. Wilson v, Jan 10 2011 *)

CROSSREFS

Cf. A008292, A180056.

Sequence in context: A197947 A220798 A220784 * A198893 A279886 A048828

Adjacent sequences:  A025582 A025583 A025584 * A025586 A025587 A025588

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 23 20:23 EST 2017. Contains 295141 sequences.