cp's OEIS Frontend

A132436 A binomial recursion: a(n) = p(n) (see comment).

%I A132436 #23 Apr 29 2023 00:05:37
%S A132436 1,1,4,20,129,1020,9542,103063,1262134,17279744,261531315,4335950346,
%T A132436 78146040374,1521220672933,31808447321848,711019048106744,
%U A132436 16919695824732249,427046133330613512,11394750238551713066,320486422239301377007,9476411014096567341034
%N A132436 A binomial recursion: a(n) = p(n) (see comment).
%C A132436 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).
%H A132436 Vaclav Kotesovec, <a href="/A132436/b132436.txt">Table of n, a(n) for n = 1..400</a>
%F A132436 Limit_{n->oo} p(n)/q(n) = (Pi-2)/(4-Pi) = 1.329896183162743847239353...
%F A132436 From _Vaclav Kotesovec_, Nov 25 2020: (Start)
%F A132436 a(n) ~ (Pi - 2) * n! / (2*sqrt(Pi*n) * log(2)^(n + 1/2)).
%F A132436 a(n) ~ (Pi - 2) * n^n / (sqrt(2) * exp(n) * log(2)^(n + 1/2)).
%F A132436 E.g.f.: 1 + x + exp(x/2)*(2*arcsin(exp(x/2)/sqrt(2)) - 1 - Pi/2) / sqrt(2 - exp(x)).
%F A132436 (End)
%t A132436 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 *)
%t A132436 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 *)
%o A132436 (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);
%Y A132436 Cf. A135147, A135148, A135149, A135150, A135074, A135075.
%K A132436 nonn
%O A132436 1,3
%A A132436 _Benoit Cloitre_, Nov 20 2007