A227464 -id:A227464 - cp's OEIS frontend

A227463 E.g.f. equals the series reversion of arcsinh(x) / exp(x).

1, 2, 10, 80, 876, 12192, 206144, 4104704, 94092112, 2440642560, 70676191840, 2260198354944, 79113937385536, 3008546200346624, 123513154739070976, 5444598073252904960, 256489070938397360384, 12859678961654923395072, 683701585124386481758720

Offset: 1

Paul D. Hanna, Jul 13 2013

nonn

Note that arcsinh(x) = log(sqrt(1+x^2) + x).

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 876*x^5/5! + 12192*x^6/6! + ...
where A( arcsinh(x)/exp(x) ) = x.

Cf. A227464.

Rest[CoefficientList[InverseSeries[Series[ArcSinh[x] / Exp[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(asinh(X)/exp(X)), n)}
for(n=1,25,print1(a(n),", "))

{a(n)=local(A=x); for(i=1,n,A=sinh(x*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

E.g.f. A(x) satisfies: A(x) = sinh(x*exp(A(x))).

a(n) ~ n^(n-1) * sqrt((1+s^2)/(1+s+s^2)) * (sqrt(1+s^2)/exp(1-s))^n, where s = 0.84184323411403778647... is the root of the equation sqrt(1+s^2)*arcsinh(s) = 1. - Vaclav Kotesovec, Jan 13 2014

A385426 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 145, 1665, 24115, 422305, 8681985, 205042625, 5471351875, 162811832625, 5345929731025, 192007183247425, 7488448738333875, 315170338129570625, 14238153926819850625, 687220571240324330625, 35293921478604240911875, 1921751625123502012140625

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Cf. A385424, A385427.

Cf. A001147, A381145, A227464, A385343.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*exp(-asin(x)))/x))

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

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

A334854 E.g.f. A(x) satisfies: A(x) = arctan(x * exp(A(x))).

Original entry on oeis.org

1, 2, 7, 32, 149, 240, -12725, -378880, -8550135, -178474240, -3550577425, -65428992000, -985800819875, -4641229465600, 548638404441075, 36708748623872000, 1716785996161588625, 69798035394846720000, 2571598453132219756375, 84313566554397999104000

Offset: 1

Ilya Gutkovskiy, May 13 2020

sign

Cf. A227464, A227466.

nmax = 20; CoefficientList[InverseSeries[Series[Tan[x]/Exp[x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest

E.g.f.: series reversion of tan(x) / exp(x).

cp's OEIS Frontend

A227463 E.g.f. equals the series reversion of arcsinh(x) / exp(x).

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A385426 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arcsin(x)) ).

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A334854 E.g.f. A(x) satisfies: A(x) = arctan(x * exp(A(x))).

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