OFFSET
1,2
COMMENTS
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
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]
Digital Library of Mathematical Functions, Permutations: Order Notation
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 DMLF link).
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
From Alois P. Heinz, Jul 21 2018: (Start)
a(n) = n * (2n-2)! * [x^(2n-2) y^(n-1)] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
a(n) = (2n)!/n [x^(2n) y^n] (1-y*x)/(1-y*exp((1-y)*x)). (End)
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
KEYWORD
nonn
AUTHOR
STATUS
approved