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A157002
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Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.
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1
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1, 0, 1, 2, 6, 17, 51, 156, 488, 1552, 5006, 16337, 53849, 179015, 599535, 2020924, 6851150, 23344138, 79902364, 274606264, 947240592, 3278404274, 11381240074, 39621423949, 138288477617, 483805404673, 1696318159457, 5959737806635
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Image of the Catalan numbers A000108 by the Riordan array (1-x,x(1-x^2)). Hankel transform is A006720(n+1).
The sequence a(n)+a(n+1) begins 1,1,3,8,23,68, ... This is A056010. The sequence a(n)+a(n-1) begins
1,1,1,3,8,23,68,... which is A025262. This is obtained by applying (1-x^2,x(1-x^2)) to the Catalan numbers.
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REFERENCES
| Guoce Xin, Proof of the Somos-4 Hankel determinants conjecture, Advances in Applied Mathematics, 42 (2009) 152-156.
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FORMULA
| G.f.: (1-sqrt(1-4x(1-x^2)))/(2x(1+x));
a(n)=sum{k=0..n, (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k)}.
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CROSSREFS
| Sequence in context: A148450 A153773 A059398 * A071717 A181665 A186239
Adjacent sequences: A156999 A157000 A157001 * A157003 A157004 A157005
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 20 2009
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