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A099363
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An inverse Chebyshev transform of 1-x.
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5
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1, -1, 1, -2, 2, -5, 5, -14, 14, -42, 42, -132, 132, -429, 429, -1430, 1430, -4862, 4862, -16796, 16796, -58786, 58786, -208012, 208012, -742900, 742900, -2674440, 2674440, -9694845, 9694845, -35357670, 35357670, -129644790, 129644790, -477638700, 477638700, -1767263190, 1767263190
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OFFSET
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0,4
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COMMENTS
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Second binomial transform of the expansion of c(-x)^3. The g.f. is transformed to 1-x under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)).
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LINKS
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FORMULA
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G.f.: (1-(1-x)c(x^2))/x where c(x) is the g.f. of the Catalan numbers A000108.
a(n) = sum{k=0..n, (k+1)C(n, (n-k)/2)(0^k-sum{j=0..k, C(k, j)(-1)^(k-j)*j})(1+(-1)^(n-k))/(n+k+2)}.
Conjecture: (n+3)*a(n) +(-n-1)*a(n-1) -4*n*a(n-2) +4*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 26 2012
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MATHEMATICA
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PROG
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(Sage)
D = [0]*(n+2); D[1] = 1
b = True; h = 2; R = []
for i in range(2*n-1) :
if b :
for k in range(h, 0, -1) : D[k] -= D[k-1]
h += 1; R.append((-1)^(h//2)*D[2])
else :
for k in range(1, h, 1) : D[k] += D[k+1]
b = not b
return R
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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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STATUS
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approved
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