A369187 The numerators of a series that converges to the Dottie Number (A003957).

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

Offset: 1

Author

Raul Prisacariu, Jan 15 2024

Keywords

• sign

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.

Examples

			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.
		

Links

• E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Crossrefs

Cf. A003957.

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.

Extensions

a(8)-a(20) from Chai Wah Wu, Feb 10 2024