OFFSET
1,3
COMMENTS
Let z(1) = x and z(n) = 1 + Sum_{k=1..n-1} (-1 + binomial(n,k))*z(k), then z(n) = p(n)*x + q(n).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..400
FORMULA
Limit_{n->oo} p(n)/q(n) = (Pi-2)/(4-Pi) = 1.329896183162743847239353...
From Vaclav Kotesovec, Nov 25 2020: (Start)
a(n) ~ (Pi - 2) * n! / (2*sqrt(Pi*n) * log(2)^(n + 1/2)).
a(n) ~ (Pi - 2) * n^n / (sqrt(2) * exp(n) * log(2)^(n + 1/2)).
E.g.f.: 1 + x + exp(x/2)*(2*arcsin(exp(x/2)/sqrt(2)) - 1 - Pi/2) / sqrt(2 - exp(x)).
(End)
MATHEMATICA
z[1] := x; z[n_] := z[n] = Expand[1 + Sum[(-1 + Binomial[n, k])*z[k], {k, 1, n-1}]]; Table[Coefficient[z[n], x], {n, 1, 30}] (* Vaclav Kotesovec, Nov 25 2020 *)
nmax = 30; Rest[Simplify[CoefficientList[Series[1 + x + E^(x/2)*(2*ArcSin[E^(x/2)/Sqrt[2]] - 1 - Pi/2)/Sqrt[2 - E^x], {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Nov 25 2020 *)
PROG
(PARI) r=1; s=-1; v=vector(120, j, x); for(n=2, 120, g=r+sum(k=1, n-1, (s+binomial(n, k))*v[k]); v[n]=g); z(n)=v[n]; p(n)=polcoeff(z(n), 1); q(n)=polcoeff(z(n), 0); a(n)=p(n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Nov 20 2007
STATUS
approved