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A100066
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Expansion of x/((1-x)sqrt(1-4x^2)).
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3
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0, 1, 1, 3, 3, 9, 9, 29, 29, 99, 99, 351, 351, 1275, 1275, 4707, 4707, 17577, 17577, 66197, 66197, 250953, 250953, 956385, 956385, 3660541, 3660541, 14061141, 14061141, 54177741, 54177741, 209295261, 209295261, 810375651, 810375651
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| An inverse Chebyshev transform of x/(1-x+x^2), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4x^2))A(xc(x^2)) where c(x) is the g.f. of the Catalan numbers A000108.
Hankel transform of a(n+1) is A120582. The Hankel transform of a(n) is -[x^n]x/(1+2x-4x^2). [From Paul Barry (pbarry(AT)wit.ie), Mar 29 2010]
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FORMULA
| a(n)=sum{k=0..n, if(mod(n-k)=2, binomial(n, (n-k)/2)*2sin(pi*k/3)/sqrt(3)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k))sin(pi*k/3)/sqrt(3)}; a(n)=sum{k=0..n, binomial(n-1, (n-1)/2)(1-(-1)^k)/2}.
a(n+1)=sum{k=0..floor(n/2), binomial(2k, k)}=sum{k=0..n, binomial(k, k/2)*(1+(-1)^k)/2};
a(2n-1) = a(2n) = A006134(n-1) = Sum[ (2k)!/(k!)^2, {k,0,n} ] for n>0. - Alexander Adamchuk (alex(AT)kolmogorov.com), Feb 23 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Mar 29 2010: (Start)
G.f.: x(1+x)/((1-x^2)*sqrt(1-4x^2))=x/((1-x)*sqrt(1-4x^2));
E.g.f.: int(exp(x-t)*Bessel_I(0,2t),t,0,x). (End)
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MAPLE
| a:=n->sum(binomial(2*j, j), j=0..n): seq(a(n/2), n=-1..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007
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CROSSREFS
| Cf. A006134.
Sequence in context: A146788 A147244 A146575 * A170832 A117783 A121445
Adjacent sequences: A100063 A100064 A100065 * A100067 A100068 A100069
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
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