%I #36 Nov 18 2023 18:16:59
%S 1,1,1,9,1126,53825,302989,2285199,133296721,9731109349,1737376806937,
%T 372236638394027,94229801087550639,27818002500902930641,
%U 591930814558449521261,9591188150350759241842,2816408483135723327055984,1394771058490469072473603553,385768133102988434073147277769
%N The numerators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.
%C 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.
%C 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)).
%H Raul Prisacariu, <a href="https://www.raulprisacariu.com/math/whittakers-root-series-going-transcendental/">Whittaker's Root Series: Going Transcendental</a>.
%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(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).
%e 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.
%e a(1) is the numerator of -(-1/2)/(1*det((1,-1/2),(1,1))=(1/2)/(3/2)=1/3
%e 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
%t 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 *)
%Y Cf. A068985 (1/e), A365594 (denominators), A323339, A323340.
%K nonn,frac
%O 1,4
%A _Raul Prisacariu_, Sep 10 2023
%E More terms from _Vaclav Kotesovec_, Oct 09 2023