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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 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] 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 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 Cf. A008292, A180056. Sequence in context: A197947 A220798 A220784 * A302657 A198893 A279886 Adjacent sequences:  A025582 A025583 A025584 * A025586 A025587 A025588 KEYWORD nonn AUTHOR STATUS approved

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Last modified October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)