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Search: id:A126120
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| 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Inverse binomial transform of A001006.
The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
Counts returning walks of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
a(n) is the coefficient of z^n in I_0(2z), where I_0 is the hyperbolic Bessel function (of the first kind) of order zero. - Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008
Essentially the same as A097331. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 15 2008
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REFERENCES
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Martin Aigner, "Catalan and other numbers: a recurrent theme", in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.
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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)=A000108(n), a(2*n+1)=0 . a(n)=A053121(n,0).
(1/Pi) Integral_{0 .. Pi } (2cos(x))^n*2sin^2(x) dx. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
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MAPLE
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with(combstruct):grammar := { BB = Sequence(Prod(a, BB, b)), a = Atom, b = Atom }: > seq(count([BB, grammar], size=n), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
BB:={E=Prod(Z, Z), S=Union(Epsilon, Prod(S, S, E))}: ZL:=[S, BB, unlabeled]: > seq(count(ZL, size=n), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007
BB:=[T, {T=Prod(Z, Z, Z, F, F), F=Sequence(B), B=Prod(F, Z, Z)}, unlabeled]: seq(count(BB, size=i), i=1..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007
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CROSSREFS
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Cf. A000108.
Sequence in context: A104035 A115333 A105523 this_sequence A090192 A097331 A094032
Adjacent sequences: A126117 A126118 A126119 this_sequence A126121 A126122 A126123
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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 06 2007
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