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A145557
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Numerators of partial sums of a certain alternating series of inverse central binomial coefficients.
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6
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1, 5, 13, 361, 31, 1193, 31021, 34467, 5273479, 1821745, 220211, 230450795, 2880634987, 1502939987, 5896829249, 12430516053889, 1381168450513, 3271188435379, 2299645470079393, 459929094015491, 819873602375609, 810854992749436603, 311867304903633289
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OFFSET
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1,2
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COMMENTS
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See A145558 for the denominators divided by 2.
The limit of the rational partial sums r(n), defined below, for n->infinity is 2*(2*phi-1)*log(phi)/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.4304089412.
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LINKS
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FORMULA
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a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n):=sum(((-1)^(k+1))/(k*binomial(2*k,k)),k=1..n).
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EXAMPLE
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Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072,...].
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MAPLE
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R:= 0;
for n from 1 to 100 do
R:= R + (-1)^(n+1)/(n*binomial(2*n, n));
a[n]:=numer(R);
od:
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PROG
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(PARI) vector(50, n, numerator(sum(k=1, n, (-1)^(k+1)/(k*binomial(2*k, k))))) \\ Michel Marcus, Oct 13 2014
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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