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A025585
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Central Eulerian numbers A(2n-1, n).
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5
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1, 4, 66, 2416, 156190, 15724248, 2275172004, 447538817472, 114890380658550, 37307713155613000, 14950368791471452636, 7246997577257618116704, 4179647109945703200884716, 2828559673553002161809327536
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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?)
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LINKS
| 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]
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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). [From Gary Detlefs, Nov 11 2011]
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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
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MATHEMATICA
| f[n_] := Sum[(-1)^j*(n - j)^(2 n - 1)*Binomial[2 n, j], {j, 0, n}]; Array[f, 14] (* From Robert G. Wilson v, Jan 10 2011 *)
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CROSSREFS
| Cf. A008292, A180056.
Sequence in context: A206500 A015475 A197947 * A198893 A048828 A003360
Adjacent sequences: A025582 A025583 A025584 * A025586 A025587 A025588
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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