A385427 E.g.f. A(x) satisfies A(x) = exp( arcsin(x * A(x)) / A(x) ).

1, 1, 1, 2, 13, 100, 861, 9536, 127737, 1938896, 33240185, 639683552, 13601898245, 316356906944, 7998251969813, 218420230243840, 6405441641302641, 200779795515236608, 6699317212660139761, 237070134772942395904, 8868209937245857514365, 349657703494298519409664

Offset: 0

Author

Seiichi Manyama, Jun 28 2025

Keywords

• nonn

Crossrefs

Cf. A385424, A385426.

Cf. A381148, A385343.

Programs

Mathematica

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 *)

PARI

a385343(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);
a(n) = sum(k=0, n, (n-k+1)^(k-1)*a385343(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A385343(n,k).

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