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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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2
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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
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OFFSET
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0,4
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COMMENTS
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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,... which 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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LINKS
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FORMULA
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G.f.: (1 - sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)).
a(n) = Sum_{k=0..n} (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k).
Conjecture: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 19 2014
0 = a(n)*(+16*a(n+1) + 16*a(n+2) - 64*a(n+3) - 42*a(n+4) + 22*a(n+5)) + a(n+1)*(+16*a(n+1) + 48*a(n+2) - 46*a(n+3) - 56*a(n+4) + 22*a(n+5)) + a(n+2)*(+32*a(n+2) + 34*a(n+3) - 8*a(n+4) - 10*a(n+5)) + a(n+3)*(+18*a(n+3) + 11*a(n+4) - 9*a(n+5)) + a(n+4)*(+3*a(n+4) + a(n+5)) for all n in Z. - Michael Somos, Feb 10 2015
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EXAMPLE
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G.f. = 1 + x^2 + 2*x^3 + 6*x^4 + 17*x^5 + 51*x^6 + 156*x^7 + 488*x^8 + ...
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[1-4x(1-x^2)])/(2x(1+x)), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)
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PROG
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(PARI) {a(n) = if( n<0, -(-1)^n / 2 * (n<-1), polcoeff( (1 - sqrt(1 - 4*x * (1 - x^2) + x^2 * O(x^n))) / (2 * x * (1 + x)), n))}; /* Michael Somos, Feb 10 2015 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1 -Sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)) )); // G. C. Greubel, Feb 26 2019
(Sage) ((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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