A365595 The numerators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

1, 1, 1, 9, 1126, 53825, 302989, 2285199, 133296721, 9731109349, 1737376806937, 372236638394027, 94229801087550639, 27818002500902930641, 591930814558449521261, 9591188150350759241842, 2816408483135723327055984, 1394771058490469072473603553, 385768133102988434073147277769

Offset: 1

Author

Raul Prisacariu, Sep 10 2023

Keywords

• nonn
• 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 1 + the Taylor expansion of log(1+x). The series obtained after applying Whittaker's Root Series Formula: 1/e-1=(-1)/1+(1/2)/(1*3/2)+(1/12)/((3/2)*(7/3))+(1/18)/((7/3)*(11/3))+(1/20)/((11/3)*(347/60))+(563/10800)/((347/60)*(3289/360))+ ... . The series can be simplified to: 1/e=1/3+1/42+1/154+9/3817+1126/1141283+ ... . The sequence is formed by the numerators of the simplified series.

The fractions in the denominators of the non-simplified series seem to be equal to terms from A323339 divided by the corresponding terms from A323340. Thus, the Whittaker's Root Series for 1 + the Taylor expansion of log(1+x) offers an alternative method for obtaining the terms of A323339 and A323340 using the determinants of Toeplitz matrices (formed using the coefficients of 1 + the Taylor expansion of log(1+x)).

Examples

			Whittaker's Root Series Formula is applied to 1 + the Taylor expansion of log(1+x) and the terms are simplified. The sequence is formed by the numerators of the simplified terms, starting with the second term in the Whittaker's Root Series.
a(1) is the numerator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
a(2) is the numerator of -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)*(7/3)) = 1/42.
		

Links

• Raul Prisacariu, Whittaker's Root Series: Going Transcendental.
• E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Crossrefs

Cf. A068985 (1/e), A365594 (denominators), A323339, A323340.

Programs

Mathematica

c[k_] := If[k < 0, 0, SeriesCoefficient[1 + Log[1 + x], {x, 0, k}]]; Table[-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, Oct 09 2023 *)

Formula

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)=1/3, c(4)=-1/4, c(n)=(1/n)*(-1)^(n+1).c(n) is simply the coefficient of x^n in the series formed by 1+ the Taylor expansion of log(1+x).

Extensions

More terms from Vaclav Kotesovec, Oct 09 2023