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A126869
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a(n) = Sum_{k, 0<=k<=n} binomial(n,floor(k/2))*(-1)^(n-k).
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14
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1, 0, 2, 0, 6, 0, 20, 0, 70, 0, 252, 0, 924, 0, 3432, 0, 12870, 0, 48620, 0, 184756, 0, 705432, 0, 2704156, 0, 10400600, 0, 40116600, 0, 155117520, 0, 601080390, 0, 2333606220, 0, 9075135300, 0, 35345263800, 0, 137846528820, 0, 538257874440, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
Contribution from Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008: (Start)
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.
Moment sequence of the trace of a random matrix in G=SO(2). If X=tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].
Also the moment sequence of the trace of the k-th power of a random matrix in USp(2)=SU(2), for all k > 2.
(End)
Contribution from Paul Barry, Aug 10 2009: (Start)
The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1-(-1)^n)/4, I=sqrt(-1).
The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1+x/sqrt(1-4x^2)) is A164111. (End)
a(n) = A204293(2*n,n): central terms of the triangle in A204293. [Reinhard Zumkeller, Jan 14 2012]
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REFERENCES
| Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, Arxiv preprint arXiv:1110.6638, 2011
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
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LINKS
| Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
| a(2*n) = binomial(2*n,n) = A000984(n), a(2*n+1)=0. a(n)=Sum_{k, 0<=k<=n}A107430(n,k)*(-1)^(n-k) = Sum_{k, 0<=k<=n}A061554(n,k)*(-1)^k.
a(n) = (1/Pi)*Integral_{t=0..Pi}cos^n(t)dt. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
E.g.f.: I_0 (2x) where I_n(x) is the modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 10 2011
G.f.: A(x)=1/sqrt(1-4*x^2) [From Vladimir Kruchinin, Apr 16 2011]
a(n) = (1/Pi)*integral(x=-2..2, x^n/sqrt((2-x)*(2+x))). [Peter Luschny, Sep 12 2011]
a(n) = (-1)^floor(n/2) Hypergeometric([-n,-n],[1], -1). [Peter Luschny, Nov 01 2011]
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EXAMPLE
| a(4)=6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.
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MATHEMATICA
| Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n, -n}, {1}, -1], {n, 0, 30}] (* Peter Luschny, Nov 01 2011 *)
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PROG
| (Haskell)
a126869 n = a204293_row (2*n) !! n -- Reinhard Zumkeller, Jan 14 2012
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CROSSREFS
| This is A000984 with interspersed zeros.
Cf. A107430, A061554, A126120.
Sequence in context: A019781 A167294 A081153 * A094233 A094659 A137437
Adjacent sequences: A126866 A126867 A126868 * A126870 A126871 A126872
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KEYWORD
| nonn
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 16 2007
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