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

%I A025585 #60 Apr 05 2024 11:07:04
%S A025585 1,4,66,2416,156190,15724248,2275172004,447538817472,114890380658550,
%T A025585 37307713155613000,14950368791471452636,7246997577257618116704,
%U A025585 4179647109945703200884716,2828559673553002161809327536,2219711218428375098854998661320
%N A025585 Central Eulerian numbers A(2n-1,n).
%C A025585 It appears to be equal to the sum over all NE lattice walks from (1,1) to (n,n) of the product over all N steps of the current x coordinate (the number of E steps which came before it plus one) times the product over all E steps of the current y coordinate. - _Jonathan Noel_, Oct 10 2018
%D A025585 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
%D A025585 B. Sturmfels, Solving Systems of Polynomial Equations, Amer. Math. Soc., 2002, see p. 27 (is that the same sequence?)
%H A025585 Alois P. Heinz, <a href="/A025585/b025585.txt">Table of n, a(n) for n = 1..200</a>
%H A025585 David H. Bailey and Jonathan M. Borwein, <a href="http://www.carmamaths.org/resources/jon/oscillatory.pdf">Experimental computation with oscillatory integrals</a>, Comtemp. Math. 517 (2010), 25-40, <a href="http://www.ams.org/mathscinet-getitem?mr=2731059">MR 2731059</a>. [Added by _N. J. A. Sloane_, Nov 02 2009]
%H A025585 Digital Library of Mathematical Functions, <a href="https://dlmf.nist.gov/26.14#T1">Permutations: Order Notation</a>
%F A025585 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 DMLF link).
%F A025585 a(n) = 2*n* A180056(n-1). - _Gary Detlefs_, Nov 11 2011
%F A025585 a(n+1)/a(n) ~ 4*n^2. - _Ran Pan_, Oct 26 2015
%F A025585 a(n) ~ sqrt(3) * 2^(2*n) * n^(2*n-1) / exp(2*n). - _Vaclav Kotesovec_, Oct 16 2016
%F A025585 From _Alois P. Heinz_, Jul 21 2018: (Start)
%F A025585 a(n) = n * (2n-2)! * [x^(2n-2) y^(n-1)] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
%F A025585 a(n) = (2n)!/n [x^(2n) y^n] (1-y*x)/(1-y*exp((1-y)*x)). (End)
%p A025585 # First program
%p A025585 A025585 := n-> add((-1)^j *(n-j)^(2*n-1) *binomial (2*n, j), j=0..n-1):
%p A025585 seq(A025585(n), n=1..30);
%p A025585 # This second program computes the list of
%p A025585 # the first m Central Eulerian numbers very efficiently
%p A025585 A025585_list :=
%p A025585    proc(m) local A, R, n, k;
%p A025585       R := 1;
%p A025585       if m > 1 then
%p A025585          A := array([seq(1,n=1..m)]);
%p A025585          for n from 2 to m do
%p A025585             for k from 2 to m do
%p A025585                A[k] := n*A[k-1] + k*A[k];
%p A025585                if n = k then R:= R, A[k] fi
%p A025585             od
%p A025585          od
%p A025585       fi;
%p A025585       R
%p A025585    end:
%p A025585 A025585_list(30); # _Peter Luschny_, Jan 11 2011
%t A025585 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 *)
%Y A025585 Cf. A008292, A180056.
%K A025585 nonn
%O A025585 1,2
%A A025585 _David W. Wilson_