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Search: id:A079489
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| A079489 |
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Series reversion of x(1-x^2)/(1+x^2)^2 expanded in odd powers of x. |
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+0
2
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| 1, 3, 22, 211, 2306, 27230, 338444, 4362627, 57788170, 781825066, 10757497972, 150073096238, 2117778107732, 30176799215196, 433586825237912, 6274885068167651, 91383942213277530, 1338275570267001458
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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G.f. A(x) satisfies xA(x^2)=(C(x)-C(-x))/(C(x)+C(-x)) where C(x) is g.f. of Catalan numbers A000108.
a(n) = ((2^(4n+2))/Gamma(1/2)) * ((Gamma(n+1/2)/(2*Gamma(n+2))) - Gamma(2n+3/2)/Gamma(2n+3)) [From David Dickson (dcmd(AT)unimelb.edu.au), Nov 10 2009]
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REFERENCES
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D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table A.1).
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FORMULA
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If x=y(1-y^2)/(1+y^2)^2 then y=x+3*x^3+22*x^5+211*x^7+2306*x^9 +...
a(n) = Sum[(-1)^k CatalanNumber[2n-k] CatalanNumber[k],{k,0,2n}]. - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006
a(n)=sum{k=0..2n, (-1)^k*A000108(k)*A000108(n-2k)}; - Paul Barry (pbarry(AT)wit.ie), Oct 09 2007
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x*(1-x^2)/(1+x^2)^2+O(x^(2*n+3))), 2*n+1))
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CROSSREFS
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Final diagonal of triangle in A078990.
Sequence in context: A098618 A006783 A001409 this_sequence A141152 A073530 A120667
Adjacent sequences: A079486 A079487 A079488 this_sequence A079490 A079491 A079492
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KEYWORD
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nonn,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jan 20 2003
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