A385424
Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).
Original entry on oeis.org
1, 1, 3, 17, 137, 1465, 19499, 311873, 5829073, 124796081, 3012319315, 80960234577, 2398138520409, 77630951407529, 2726829925494011, 103300796618253825, 4198494172961579169, 182239547736082960737, 8414068749731088539299, 411754575622058760824593
Offset: 0
A385427
E.g.f. A(x) satisfies A(x) = exp( arcsin(x * A(x)) / A(x) ).
Original entry on oeis.org
1, 1, 1, 2, 13, 100, 861, 9536, 127737, 1938896, 33240185, 639683552, 13601898245, 316356906944, 7998251969813, 218420230243840, 6405441641302641, 200779795515236608, 6699317212660139761, 237070134772942395904, 8868209937245857514365, 349657703494298519409664
Offset: 0
-
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 *)
-
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));
A385501
Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arctanh(x)) ).
Original entry on oeis.org
1, 1, 3, 18, 165, 2040, 31815, 599760, 13268745, 337115520, 9674678475, 309554784000, 10927053262125, 421849524096000, 17682153623909775, 799730490214656000, 38820939579369572625, 2013202580708487168000, 111081054630965602057875, 6497703571257963896832000
Offset: 0
-
nmax=20; CoefficientList[(1/x) *InverseSeries[Series[x * Exp[-ArcTanh[x]],{x,0,nmax}],x] ,x]Range[0,nmax-1]! (* Stefano Spezia, Jul 01 2025 *)
-
a(n) = n!*sum(k=0, n, binomial(n, k)*binomial(n/2+k+1/2, n)/(n+2*k+1));
Showing 1-3 of 3 results.