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A157125
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A transform of the Catalan numbers.
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2
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1, -1, -1, 0, 2, 1, -1, -4, -2, 4, 12, 4, -20, -39, 3, 92, 118, -84, -388, -308, 596, 1528, 508, -3292, -5556, 1154, 16034, 17940, -18160, -71243, -45913, 127124, 290278, 46128, -710864, -1067564, 485108, 3504680, 3362756, -4957812, -15669148
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1-x)*(sqrt(1+x^2+4*x^3)-sqrt(1+x^2))/(2*x^3*sqrt(1+x^2));
a(n)=sum{k=0..n, (-1)^C(n-k+1,2)*C(floor((n-k)/2)),k)*A000108(k)}.
Conjecture: (n+3)*(n-2)*a(n) -4*a(n-1) +2*(n^2-n-4)*a(n-2) +2*(2*n^2-7*n+2)*a(n-3) +(n+1)*(n-4)*a(n-4) +2*(n-1)*(2*n-7)*a(n-5)=0. - R. J. Mathar, Nov 15 2012
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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Divisor x^3 inserted in the g.f. - R. J. Mathar, Feb 06 2015
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STATUS
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approved
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