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%I #25 Aug 17 2018 19:47:21
%S 3,1,3,10,39,184,1047,7000,53571,460936,4404603,46296040,530878719,
%T 6595091944,88232942847,1264741738120,19337532032091,314144393039176,
%U 5403576523773603,98110258621524520,1875097757416854999,37629001852534817704,791088129700026499047
%N Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k) with a(0)=3, a(1)=1.
%C One of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064.
%H Stanislav Sykora, <a href="/A289066/b289066.txt">Table of n, a(n) for n = 0..200</a>
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL6Math17.001">Sequences related to the differential equation f'' = af'f</a>, Stan's Library, Vol. VI, Jun 2017.
%F E.g.f.: -sqrt(7)/tanh(z*sqrt(7)/2 - arccosh(3/sqrt(2))).
%F E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(7)/tanh(z*sqrt(7)/2 + arccosh(3/sqrt(2))).
%F a(n) ~ 2 * n! * 7^((n+1)/2) / log(8 + 3*sqrt(7))^(n+1). - _Vaclav Kotesovec_, Jun 24 2017
%p f:= proc(n) option remember; add(binomial(n-2,k)*procname(k)*procname(n-1-k),k=0..n-2) end proc:
%p f(0):= 3: f(1):= 1:
%p map(f, [$0..50]); # _Robert Israel_, Jul 20 2017
%t a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 3; a[1] = 1; Array[a, 23, 0] (* _Jean-François Alcover_, Jul 20 2017 *)
%o (PARI) c0=3; c1=1; nmax = 200; a = vector(nmax+1); a[1]=c0; a[2]=c1; for(m=0,#a-3,a[m+3]=sum(k=0,m,binomial(m,k)*a[k+1]*a[m+2-k])); a
%Y Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3).
%K nonn
%O 0,1
%A _Stanislav Sykora_, Jun 23 2017
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