cp's OEIS Frontend

A209305 Exponential generating function A(x) is the unique solution of the Cauchy problem: A'(x) = exp(x+A(x)^2-1), A(0) = 1.

%I A209305 #25 May 14 2025 09:08:39
%S A209305 1,1,3,17,151,1821,27675,507177,10879247,267329317,7403007507,
%T A209305 228096010961,7739098803559,286704779800173,11514881722287051,
%U A209305 498352218891144249,23120905868853862655,1144719051552552433525,60241163501500355522499
%N A209305 Exponential generating function A(x) is the unique solution of the Cauchy problem: A'(x) = exp(x+A(x)^2-1), A(0) = 1.
%H A209305 G. C. Greubel, <a href="/A209305/b209305.txt">Table of n, a(n) for n = 0..375</a>
%F A209305 E.g.f.: A(x) = inverse_erf((2*exp(x)-2+exp(1)*sqrt(Pi)*erf(1)) / (exp(1)*sqrt(Pi))), where inverse_erf is the inverse of the error function.
%F A209305 a(n) = a(n-1) + 2*Sum_{k=0..n-2} C(n-2,k)*a(k)*b(n-2-k) for n>0, a(0) = 1, where b(n) = Sum_{k=0..n} C(n,k)*a(k+1)*a(n-k+1) (A209306).
%F A209305 Limit_{n->oo} (a(n)/n!)^(1/n) = 1 / log(1 + exp(1)*sqrt(Pi)*erfc(1)/2) = 3.112237454247335904679793089483819785... . - _Vaclav Kotesovec_, Mar 31 2017
%t A209305 (* Expansion of the generating series *)
%t A209305 A[x_] := InverseErf[(2 Exp[x] - 2 + Exp[1] Sqrt[Pi] Erf[1])/(Exp[1] Sqrt[Pi])];
%t A209305 CoefficientList[Series[A[x], {x, 0,40}], x] Table[n!, {n, 0,40}]
%t A209305 (* Recurrences - More efficient *)
%t A209305 a[n_] := a[n] = a[n-1]+2Sum[Binomial[n-2,k]a[k]b[n-2-k],{k,0,n-2}];
%t A209305 a[1] = 1;
%t A209305 a[0] = 1;
%t A209305 b[n_] := Sum[Binomial[n, k]a[k + 1]a[n - k + 1], {k, 0, n}];
%t A209305 Table[a[n], {n, 0, 100}]
%Y A209305 Cf. A026944, A209306, A209307.
%K A209305 nonn
%O A209305 0,3
%A A209305 _Emanuele Munarini_, Jan 18 2013