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 A201731 #15 Feb 16 2025 21:22:34
%S A201731 1,4,54,1544,75750,5676492,603041334,86210654224,15958892198070,
%T A201731 3713676157320020,1061084890984465446,365202873520507832856,
%U A201731 149027843082185351506950,71144948740332156241755868,39282974873393643411310747350,24840594864924259316810487005216
%N A201731 a(n) = [x^n/n!] log( (n - (n-1)*exp(x)) / (n+1 - n*exp(x)) ).
%C A201731 The function log((n - (n-1)*exp(x))/(n+1 - n*exp(x))) equals the n-th iteration of log(1/(2-exp(x))), the e.g.f. of A000629 (with offset 1), where A000629(n) is the number of necklaces of partitions of n+1 labeled beads.
%F A201731 a(n) = (n+1) * A201732(n+1).
%e A201731 Let G(x) = log(1/(2-exp(x))) then the coefficients of x^n/n! in the k-th iteration of G(x) begin:
%e A201731 k=1: [(1), 2, 6, 26, 150, 1082, 9366, 94586, ..., A000629(n-1), ...];
%e A201731 k=2: [1,(4), 24, 196, 2040, 25924, 390264, 6804676, ..., A201338(n), ...];
%e A201731 k=3: [1, 6,(54), 654, 9990, 184686, 4015494, 100531374, ...];
%e A201731 k=4: [1, 8, 96, (1544), 31200, 760328, 21721056, 712459784, ...];
%e A201731 k=5: [1, 10, 150, 3010, (75750), 2295010, 81378150, 3307983010, ...];
%e A201731 k=6: [1, 12, 216, 5196, 156600, (5676492), 240593976, 11679764556, ...];
%e A201731 k=7: [1, 14, 294, 8246, 289590, 12224534, (603041334), 34053651926, ...];
%e A201731 k=8: [1, 16, 384, 12304, 493440, 23777296, 1338417024, (86210654224), ...]; ...
%e A201731 where the coefficients in parenthesis form the initial terms of this sequence.
%o A201731 (PARI) {a(n)=n!*polcoeff(log((n - (n-1)*exp(x+x*O(x^n)))/(n+1 - n*exp(x+x*O(x^n)))),n)}
%Y A201731 Cf. A000629, A201338, A201732.
%K A201731 nonn
%O A201731 1,2
%A A201731 _Paul D. Hanna_, Dec 04 2011