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A126869 a(n) = Sum_{k, 0<=k<=n} binomial(n,floor(k/2))*(-1)^(n-k). 19
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; text; internal format)
OFFSET

0,3

COMMENTS

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

From Andrew V. Sutherland, 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)

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]

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

LINKS

Table of n, a(n) for n=0..43.

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

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

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462

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, 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) [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]

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

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

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

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

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

G.f.: G(0)/(1+x), where G(k) = 1 + x*(2+5*x)*(4*k+1)/((4*k+2)*(1+x)^2 - 2*(2*k+1)*(4*k+3)*x*(2+5*x)*(1+x)^2/((4*k+3)*x*(2+5*x) + 4*(k+1)*(1+x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014

EXAMPLE

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

MAPLE

seq((-1)^(n/2)*pochhammer(-n, n/2)/(n/2)!, n=0..43); # Peter Luschny, May 17 2013

MATHEMATICA

Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n, -n}, {1}, -1], {n, 0, 30}] (* Peter Luschny, Nov 01 2011 *)

PROG

(Haskell)

a126869 n = a204293_row (2*n) !! n  -- Reinhard Zumkeller, Jan 14 2012

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

KEYWORD

nonn

AUTHOR

Philippe Deléham, Mar 16 2007

STATUS

approved

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Last modified April 19 03:31 EDT 2014. Contains 240738 sequences.