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A025585 Central Eulerian numbers A(2n-1,n). 6
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

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Last modified March 26 23:19 EDT 2017. Contains 284141 sequences.