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 A126869 a(n) = Sum_{k, 0<=k<=n} binomial(n,floor(k/2))*(-1)^(n-k). 15

%I

%S 1,0,2,0,6,0,20,0,70,0,252,0,924,0,3432,0,12870,0,48620,0,184756,0,

%T 705432,0,2704156,0,10400600,0,40116600,0,155117520,0,601080390,0,

%U 2333606220,0,9075135300,0,35345263800,0,137846528820,0,538257874440,0

%N a(n) = Sum_{k, 0<=k<=n} binomial(n,floor(k/2))*(-1)^(n-k).

%C Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.

%C From Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008: (Start)

%C Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.

%C 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].

%C 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.

%C (End)

%C From _Paul Barry_, Aug 10 2009: (Start)

%C 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).

%C The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1+x/sqrt(1-4x^2)) is A164111. (End)

%C a(n) = A204293(2*n,n): central terms of the triangle in A204293. [_Reinhard Zumkeller_, Jan 14 2012]

%C a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - _Wolfdieter Lang_, Oct 10 2012

%D 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

%D Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, Arxiv preprint arXiv:1203.1476, 2012. - From _N. J. A. Sloane_, Sep 14 2012

%D Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>.

%F 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.

%F a(n) = (1/Pi)*Integral_{t=0..Pi}cos^n(t)dt. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008

%F 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

%F G.f.: A(x)=1/sqrt(1-4*x^2) [_Vladimir Kruchinin_, Apr 16 2011]

%F a(n) = (1/Pi)*integral(x=-2..2, x^n/sqrt((2-x)*(2+x))). [_Peter Luschny_, Sep 12 2011]

%F a(n) = (-1)^floor(n/2) Hypergeometric([-n,-n],[1], -1). [_Peter Luschny_, Nov 01 2011]

%F n*a(n) +(n-1)*a(n-1) +4*(-n+1)*a(n-2) +4*(-n+2)*a(n-3)=0. - _R. J. Mathar_, Dec 03 2012

%F E.g.f.: E(0)/(1-x) where E(k) = 1 - x/(1 - x/(x - (k+1)^2/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Apr 05 2013

%F E.g.f.: 1 + x^2/(Q(0) - x^2), where Q(k)= x^2 + (k+1)^2 - x^2*(k+1)^2/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 28 2013

%F G.f.: 1/(1 - 2*x^2*Q(0)), where Q(k)= 1 + (4*k+1)*x^2/(k+1 - x^2*(2*k+2)*(4*k+3)/(2*x^2*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013

%e a(4)=6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.

%p seq((-1)^(n/2)*pochhammer(-n,n/2)/(n/2)!, n=0..43); # _Peter Luschny_, May 17 2013

%t Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n,-n},{1},-1],{n,0,30}] (* _Peter Luschny_, Nov 01 2011 *)

%o a126869 n = a204293_row (2*n) !! n -- _Reinhard Zumkeller_, Jan 14 2012

%Y This is A000984 with interspersed zeros.

%Y Cf. A107430, A061554, A126120.

%K nonn,changed

%O 0,3

%A _Philippe DELEHAM_, Mar 16 2007

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