A291482 Expansion of e.g.f. arcsin(x)*exp(x).

0, 1, 2, 4, 8, 24, 80, 456, 2368, 20352, 139648, 1577984, 13327360, 185992832, 1860708096, 30882985472, 356724338688, 6860887896064, 89815091306496, 1963843714723840, 28724760194564096, 703639672161697792, 11370790299166343168, 308435832182144040960, 5456591088206554333184, 162354575283061816197120

Offset: 0

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

• nonn

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 24*x^5/5! + ...
		

Crossrefs

Cf. A001818, A009545, A012316, A081919 (first differences).

Programs

Maple

a:=series(arcsin(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSin[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Exp[x] x Sqrt[1 - x^2]/(1 + ContinuedFractionK[-2 x^2 Floor[(k + 1)/2] (2 Floor[(k + 1)/2] - 1), 2 k + 1, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Sum[(x^(2 k + 1) Pochhammer[1/2, k])/(k! + 2 k k!), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n,2k+1]Binomial[2k,k] (2k)!/4^k,{k,0,(n-1)/2}],{n,0,12}] (* Emanuele Munarini, Dec 17 2017 *)

Maxima

makelist(sum(binomial(n,2*k+1)*binomial(2*k,k)*(2*k)!/4^k,k,0,floor((n-1)/2)),n,0,12); /* Emanuele Munarini, Dec 17 2017 */

PARI

x='x+O('x^99); concat(0, Vec(serlaplace(asin(x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

Formula

E.g.f.: exp(x)*x*sqrt(1 - x^2)/(1 - 1*2*x^2/(3 - 1*2*x^2/(5 - 3*4*x^2/(7 - 3*4*x^2/(9 - ...))))), a continued fraction.

a(n) ~ (exp(2) - (-1)^n) * n^(n-1) / exp(n+1). - Vaclav Kotesovec, Aug 26 2017

From Emanuele Munarini, Dec 17 2017: (Start)

a(n) = Sum_{k=0..(n-1)/2} binomial(n,2*k+1)*binomial(2*k,k)* (2k)!/4^k.

a(n+4) - 2*a(n+3) - (n^2+4*n+3)*a(n+2) + (n+2)*(2*n+3)*a(n+1) - (n+1)*(n+2)*a(n) = 0. (End)