

A369187


The numerators of a series that converges to the Dottie Number (A003957).


0



1, 1, 1, 3, 1, 205, 4439, 111021, 1724351, 2074717, 2567577481, 246042951203, 14444487376705, 726562139423955, 1473171168838825, 1178164765176836393, 204468301714665778099, 138848947223110087743421, 11701779801284441802592247, 7774256876827576332115737
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OFFSET

1,4


COMMENTS

Whittaker's root series formula is applied to 1  x  x^2/2! + x^4/4!  x^6/6! + ..., which is the Taylor expansion of cos(x)  x. The following infinite series for the Dottie number (D) is obtained: D = 1/1  1/3 + 1/12  3/260 + 1/5720 + 205/314248  4439/17255072 ... . The sequence is formed by the numerators of the series.


LINKS



FORMULA

a(1) = 1; for n > 1, a(n) is the numerator of the simplified fraction det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=1, c(2)=1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)x.


EXAMPLE

a(1) is the numerator of 1/1 = 1/1.
a(2) is the numerator of simplified (1/2!)/(1* det ToeplitzMatrix((1,1),(1,1/2!))) = (1/2)/(3/2) = 1/3.
a(3) is the numerator of the simplified det ToeplitzMatrix((1/2!,1),(1/2!,0))/(det ToeplitzMatrix((1,1),(1,1/2!))*det ToeplitzMatrix((1,1,0),(1,1/2!,0)) = (1/4)/((3/2)*2) = 1/12.


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KEYWORD

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EXTENSIONS



STATUS

approved



