A367597 The numerators of a series that converges to log(2) obtained using Whittaker's Root Series Formula.

1, -1, 1, 1, -7, -13, 115, 5657, -50759, -2129735, 14303129, 567824371, -668626477, -2536766233351, -22685437440167, 6386546752479769, 164094712558603577, -8391397652598907629, -441809873725219477581, 29040435128150468206405, 2814195975064870847100773

Offset: 1

Author

Raul Prisacariu, Nov 24 2023

Keywords

• sign
• frac

Comments

The Whittaker's root series formula is applied to -1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + ..., which is the Taylor expansion of e^x with the first coefficient having a negative sign (-1 instead of 1). We obtain log(2) = 1 - 1/3 + 1/39 + 1/975 - 7/40575 - 13/844501 + 115/73824373 + 5657/25814174655 .... 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 -(-1)^2*(1/2!)/(1*det((1,1/2!),(-1,1)))=-(1/2)/(1*(3/2))=-1/3
a(3) is the numerator of -(-1)^3*det((1/2!,1/3!),(1,1/2!))/(det((1,1/2!),(-1,1))*det((1,1/2!,1/3!),(-1,1,1/2!),(0,-1,1)))=(1/12)/((3/2)*(13/6))=1/39
		

Links

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

Crossrefs

Cf. A002162, A365595, A367596 (denominators).

Programs

Mathematica

c[k_] := If[k < 0, 0, SeriesCoefficient[Exp[x] - 2, {x, 0, k}]]; Join[{1}, Table[(-1)^n*Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Numerator] (* Vaclav Kotesovec, Nov 26 2023 *)

Formula

a(n) is the numerator of the simplified fraction -(-1)^n*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)=1/3!, c(4)=1/4!,c(n)=1/n!.

Extensions

More terms from Vaclav Kotesovec, Nov 26 2023