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A053987
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Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).
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5
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1, 6, 59, 820, 14701, 322602, 8372951, 250865928, 8521068601, 323549740910, 13580568049619, 624382580541564, 31205548459028581, 1684475234207001810, 97668358035547076399, 6053753722969711734928, 399450077357965427428849, 27955451661334610208284502
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k*(2*n-2*k-1)!/((n-2*k-1)! * (2*k+1)!). - Benoit Cloitre, Jan 03 2006
a(n+1) = (4*n+2)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*4^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1, k)*binomial(n-k-1/2, k+1/2), see A058798. (End)
a(n) = 4^n*Gamma(n+1/2)*hypergeometric([1/2-n/2,1-n/2], [3/2,1/2-n,1-n], -1/4)/sqrt(4*Pi). - Peter Luschny, Sep 10 2014
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MAPLE
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A053987 := n -> local k; add((-1)^k*(2*n-2*k-1)!/((n-2*k-1)!*(2*k+1)!), k = 0..floor((n-1)/2)); seq(A053987(n), n = 1..20); # G. C. Greubel, May 17 2020
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MATHEMATICA
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Rest[CoefficientList[Series[Sin[(1-Sqrt[1-4*x])/2]/Sqrt[1-4*x], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 25 2014 *)
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PROG
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(PARI) a(n)=sum(k=0, floor((n-1)/2), (-1)^k*(2*n-2*k-1)!/(n-2*k-1)!/(2*k+1)!) \\ Benoit Cloitre, Jan 03 2006
(Sage)
return 4^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], -1/4)/sqrt(4*pi)
(Magma)
A053987:= func< n| &+[(-1)^k*Factorial(2*n-2*k-1)/(Factorial(n-2*k-1)* Factorial(2*k+1)): k in [0..Floor((n-1)/2)]] >;
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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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EXTENSIONS
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STATUS
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approved
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