%I #35 Jun 12 2020 04:21:32

%S 1,2,5,19,99,653,5187,48117,510271,6088301,80716427,1177130893,

%T 18727404639,322769897573,5990916997611,119139798166245,

%U 2527255556219167,56960055683893853,1359299747696197931,34240584053654816797,907911436336049691519,25277557586594907583733,737276033151104902965963

%N a(n+1) = 1 + Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) for n >= 0 with a(0) = 1.

%H Seiichi Manyama, <a href="/A054687/b054687.txt">Table of n, a(n) for n = 0..429</a>

%F a(n) ~ n! / r^(n+1), where r = 0.7542714558461742549095127982603266798910769877... is the smallest positive real root of the equation (BesselJ(1, 2) - BesselJ(0, 2))*BesselY(0, 2*exp(r/2)) + BesselJ(0, 2*exp(r/2))*(BesselY(0, 2) - BesselY(1, 2)) = 0. - _Vaclav Kotesovec_, Mar 02 2014, updated Apr 26 2020

%F E.g.f. A(x) satisfies A'(x) = exp(x) + A(x)^2 with A(0) = 1. - _Petros Hadjicostas_, Apr 25 2020

%F E.g.f.: exp(x/2)*(BesselJ(2, 2)*BesselY(1, 2*exp(x/2)) - BesselJ(1, 2*exp(x/2)) * BesselY(2, 2)) / (BesselJ(2, 2)*BesselY(0, 2*exp(x/2)) - BesselJ(0, 2*exp(x/2)) * BesselY(2, 2)). - _Vaclav Kotesovec_, Apr 26 2020

%e a(5) = 1 + 1*a(0)*a(4) + 4*a(1)*a(3) + 6*a(2)*a(2) + 4*a(3)*a(1) + 1*a(4)*a(0) = 1 + 1*1*99 + 4*2*19 + 6*5*5 + 4*19*2 + 1*99*1 = 653.

%t nmax=20; b = ConstantArray[0,nmax+2]; b[[1]]=1; Do[b[[n+2]] = 1 + Sum[Binomial[n,k]*b[[k+1]]*b[[n-k+1]],{k,0,n}],{n,0,nmax}]; b (* _Vaclav Kotesovec_, Mar 02 2014 *)

%o (PARI) lista(nn)={my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n]= 1 + sum(k=0, n-2, binomial(n-2, k)*a[k+1]*a[n-k-1])); for(n=1, nn, print1(a[n], ","))}; \\ _Petros Hadjicostas_, Jun 11 2020

%Y Cf. A052886.

%K easy,nonn

%O 0,2

%A _Leroy Quet_, Apr 19 2000

%E More terms from _James A. Sellers_, Apr 20 2000