A369187 - OEIS

%I #20 Jan 08 2025 11:43:53

%S 1,-1,1,-3,1,205,-4439,111021,-1724351,2074717,2567577481,

%T -246042951203,14444487376705,-726562139423955,1473171168838825,

%U 1178164765176836393,-204468301714665778099,138848947223110087743421,-11701779801284441802592247,7774256876827576332115737

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

%C 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.

%H E. T. Whittaker and G. Robinson, <a href="https://archive.org/details/calculusofobserv031400mbp/page/n139/mode/2up">The Calculus of Observations</a>, London: Blackie & Son, Ltd. 1924, pp. 120-123.

%F 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.

%e a(1) is the numerator of -1/-1 = 1/1.

%e a(2) is the numerator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.

%e 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.

%Y Cf. A003957.

%K sign

%O 1,4

%A _Raul Prisacariu_, Jan 15 2024

%E a(8)-a(20) from _Chai Wah Wu_, Feb 10 2024