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 A385427 #12 Jul 05 2025 05:05:11
%S A385427 1,1,1,2,13,100,861,9536,127737,1938896,33240185,639683552,
%T A385427 13601898245,316356906944,7998251969813,218420230243840,
%U A385427 6405441641302641,200779795515236608,6699317212660139761,237070134772942395904,8868209937245857514365,349657703494298519409664
%N A385427 E.g.f. A(x) satisfies A(x) = exp( arcsin(x * A(x)) / A(x) ).
%F A385427 a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A385343(n,k).
%F A385427 a(n) ~ s*(1 - r^2*s^2)^(3/4) * n^(n-1) / (sqrt(r^2*s^2*(2 + r*sqrt(1 - r^2*s^2) - r^2*s^2) - 1) * exp(n) * r^(n - 1/2)), where r = 0.4947196925654744939290429342422921705036054462455... and s = 1.929162378596122962197524561455700427559144822670... are the roots of the system of equations exp(arcsin(r*s)/s) = s, r*s/sqrt(1 - r^2*s^2) - arcsin(r*s) = s. - _Vaclav Kotesovec_, Jul 05 2025
%t A385427 nmax = 20; A[_] = 1; Do[A[x_] = E^(ArcSin[x*A[x]]/A[x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 05 2025 *)
%o A385427 (PARI) a385343(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);
%o A385427 a(n) = sum(k=0, n, (n-k+1)^(k-1)*a385343(n, k));
%Y A385427 Cf. A385424, A385426.
%Y A385427 Cf. A381148, A385343.
%K A385427 nonn
%O A385427 0,4
%A A385427 _Seiichi Manyama_, Jun 28 2025