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URL: https://oeis.org/A053987

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A053987
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-...))))))).
6
1, 6, 59, 820, 14701, 322602, 8372951, 250865928, 8521068601, 323549740910, 13580568049619, 624382580541564, 31205548459028581, 1684475234207001810, 97668358035547076399, 6053753722969711734928, 399450077357965427428849, 27955451661334610208284502
OFFSET
1,2
LINKS
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
FORMULA
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
E.g.f.: 1-cos(x*C(x)), C(x)=(1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
From Peter Bala, Aug 01 2013, (Start)
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) ~ sin(1/2) * 2^(2*n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Feb 25 2014
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
(SageMath)
def A053987(n):
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)
[round(A053987(n).n(100)) for n in (1..18)] # Peter Luschny, Sep 10 2014
(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)]] >;
[A053987(n) : n in [1..20]]; // G. C. Greubel, May 17 2020
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Vladeta Jovovic, Apr 03 2000
EXTENSIONS
a(16)-a(17) from Wesley Ivan Hurt, Feb 28 2014
STATUS
approved