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Search: id:A126869
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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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9
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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; listen)
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OFFSET
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0,3
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COMMENT
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Hankel transform is 2^n . Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
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]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
Also the moment sequence of the trace of the kth power of a random matrix in USp(2)=SU(2), for all k > 2. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
Contribution from Paul Barry (pbarry(AT)wit.ie), 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)
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REFERENCES
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Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
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LINKS
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Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
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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
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CROSSREFS
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Cf. A107430, A061554.
Cf. 126120.
Sequence in context: A019781 A167294 A081153 this_sequence A094233 A094659 A137437
Adjacent sequences: A126866 A126867 A126868 this_sequence A126870 A126871 A126872
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KEYWORD
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nonn
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 16 2007
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