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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-...))))))). 3
1, 6, 59, 820, 14701, 322602, 8372951, 250865928, 8521068601, 323549740910, 13580568049619, 624382580541564, 31205548459028581, 1684475234207001810, 97668358035547076399, 6053753722969711734928, 399450077357965427428849, 27955451661334610208284502 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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

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

(Sage)

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

CROSSREFS

Cf. A053988, A058798, A001517.

Sequence in context: A089153 A075136 A024382 * A024270 A024271 A271964

Adjacent sequences:  A053984 A053985 A053986 * A053988 A053989 A053990

KEYWORD

nonn,frac,easy

AUTHOR

Vladeta Jovovic, Apr 03 2000

EXTENSIONS

a(16)-a(17) from Wesley Ivan Hurt, Feb 28 2014

STATUS

approved

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Last modified January 22 22:16 EST 2020. Contains 331166 sequences. (Running on oeis4.)