This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365594 #48 Apr 13 2025 07:11:29
%S A365594 3,42,154,3817,1141283,119706444,1396550916,20958700652,2359646218028,
%T A365594 324742403298918,107268957934572210,41877140987048387615,
%U A365594 19073758392921536694655,10024177256513161424322680,376301673554116445531842536,10673126660749797308728534491
%N A365594 The denominators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.
%C A365594 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 denominators of the simplified series.
%C A365594 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 A365594 Raul Prisacariu, <a href="https://www.raulprisacariu.com/math/whittakers-root-series-going-transcendental/">Whittaker's Root Series: Going Transcendental</a>.
%H A365594 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 A365594 a(n) is the denominator 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 A365594 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 denominators of the simplified terms, starting with the second term in the Whittaker's Root Series.
%e A365594 a(1) is the denominator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
%e A365594 a(2) is the denominator 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 A365594 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}] // Denominator (* _Vaclav Kotesovec_, Oct 09 2023 *)
%Y A365594 Cf. A068985 (1/e), A365595 (numerators), A323339, A323340.
%K A365594 nonn,frac
%O A365594 1,1
%A A365594 _Raul Prisacariu_, Sep 10 2023
%E A365594 a(6)-a(7) corrected and extended by _Vaclav Kotesovec_, Oct 09 2023